Crusader Kings III

For this achievement, we need to convert all counties on the Iberian peninsula plus the three islands duchy of Mallorca to the Mozarabic faith, which even taken by itself will take quite some time, so expect an at least medium length game.

1) Which other major achievements should we go for in the same campaign

We don't technically HAVE to end the Iberian struggle to get this achievement, but it would be silly not to do so, since some struggle phases slow down conversion speed and all that infighting certainly makes converting all of Iberia to one faith more difficult. Also you need to end the struggle to be able to found the Empire of Hispania, which - again - is not mandatory for the achievement, but it helps a lot to have a stable unified realm.

You can combine this achievement either with the "Iberian Hostilities" or "Iberian Conciliation" achievement, depending on which struggle ending you go for. Iberian Hostilities is slightly more helpful, since it can give you a permanent bonus to conversion speed at the cost of different faith opinion and it doesn't restrict religious wars, but Iberian Conciliation also works well, since it makes it pretty easy to unite Iberia by mostly diplomatic means afterwards and while it restricts holy wars, it doesn't restrict county conversion. Just don't go for Iberian compromise, since that one permanently prevents creating a unified Empire of Hispania and turns all existing small kingdoms into miniature Empire titles. It's not impossible to rule and convert all of Iberia after that, but it sure gets a whole lot more complicated.
The achievement description isn't clear on whether you also have to rule all of Iberia (like it is required for the Reconquista achievement even though the description doesn't say so) or whether you can have remaining independent rulers in Iberia, as long as those rulers and their counties are Mozarabic. Aim at conquering all of Iberia including fellow Mozarabic rulers just in case.

Aside from one of those ending decision achievements, you can also do the "sibling rivalry" achievement if you play as one of the Jimena siblings in 1066 or the "holidaying in Iberia" achievement if you start as a character from outside Iberia, then become an involved character and end the struggle.

Unlike the Reconquista or the Al Andalus achievment, Andalusian inquisition doesn't require you to start as a Mozarab character, so you technically can do one of those other two first and then convert to Mozarab faith after you have the Reconquista or Al Andalus achievement and proceed to convert all of Iberia to Mozarabism in the same playthrough. This will of course take you longer than going straight for spreading Mozarabism, but it will still be much shorter than doing three seperate campaigns for each of the three Iberian faith domination achievements.

2) Which starting character to pick?

Best starting characters may depend on which other achievements you are going for (see 1). Other than that, it really doesn't matter much who your starting character is and what he can do (the more powerful they start, the easier the campaign will be, obviously. Having good diplomacy and learning helps with asking vassals to convert, but your stats are irrelevant for county conversion).
If you want the campaign to be as fast as possible, consider to start NOT AS A MOZARAB, but as a Muwalladi ruler. Even if you don't want to aim for the Al Andalus achievement first, it speeds up the county conversion a lot if you start as Muwalladi, conquer as much of your fellow Muwalladi counties as possible and then convert to Mozarabism, once you are established and well liked enough that (almost) all of your vassals voluntarily convert with you. This comes with a good chance for the poluplation of same faith domain counties of those voluntarily converting rulers to instantly convert with them (I believe somewhere around 30%, might be modified by popular opinion), which saves you a whole lot of time of having your court chaplain doing the convert county task. I recommend starting as Muwalladi because they take especially long to convert the regular way because of their pluralistic and adaptive views. Of course you also could pull off the same trick with Catholic or A'sharia, but doing it with the Muwalladi is the most useful option.

3) What to do?

For general tips on how to end the Iberian struggle and conquer Iberia, look under the respective struggle or conquest achievements, I'll only give specific tips on conversion and on dealing with the optional Mozarabic reformation decision here, in order to not overload this guide and be repetitive.

3.1) Converting or getting rid of different faith vassals

Due to the pluralist nature of Mozarabism (unless you changed that, see 3.3), you won't automatically dethrone different faith rulers upon conquest, so you'll likely end up with some different faith vassals during your conquest. Those will have to convert or have to go ASAP, because you converting counties causes constant strife with those minority vassals and they may counteract your efforts by converting their own domain back to their faith. Their domain population will not convert with them if they convert on request of their liege, but still asking them to convert is often the smoother way to do it than revoking their titles, especially since being of a different faith is not in itself a title revocation reason for a pluralist faith.

Your chances of your vassal accepting your demand to convert depend on your learning score vs their learning score, their opinion of you (being a friend or lover comes with an additional boost to the chance on top of that) and certain character traits (like e.g. stubborn or arrogant). Zealous characters are very hard to convert, but it isn't impossible. Cynical characters are easier to convert, but you might still prefer to get rid of cynical vassals, since they are very unlikely to use the convert county councillor task themselves after their conversion. Always try to get the acceptance chance to 100% or close before you actually ask them, because if they refuse, there will be a very long cooldown during which you can't ask them again.

A weak hook on your vassal can help you boost the acceptance chance (which may be enough to convert a zealot who likes you a lot), but doesn't cause a guaranteed success with a vassal that is otherwise very reluctant to convert. If you have a strong hook on them, you can force them. You can also often force them if you have them in prison for some reason (justified or unjustified arrest, capture during rebellion, abduction), but some zealous characters rather stay in prison than convert.

If you can't get them to convert, you'll have to revoke their titles. If you don't want to do that for some reason, you can also play the long game and offer them a Mozarab guardian for their underage heir who is tasked to convert the child's faith (then optionally have your assassins speed up the succession).

3.2) Converting counties

Converting counties is done by having your court chaplain do the convert county councillor task and will usually take several years per county. The learning score of your court chaplain has a huge impact on the duration, so since you can't fire an inept bishop, consider murdering them until you get one with a decent learning score (at least 12, the more the better). Fortunately, you don't have to be a master of intrigue for that, because it's very easy to find agents in your own court for this who will boost the chance up to 95% even if your own intrigue skill is only mediocre.

If you have a decent and not too old bishop, you can boost their learning score a bit by going on pilgrim journeys, which will usually cause him to tag along and earn the pilgrim and traveller traits. It can also help to make them your court physician, especially if you do the learning lifestyle (which has a chance for events in which you help your court physician translating medical texts of old philosophers, permanently increasing their learning score by 2 on success, can happen multiple times).

Your own learning score doesn't have direct impact on conversion speed, but do pick up the perks "zealous prosyletizer" and "icon of faith" from the theology skill tree. The first one increases conversion speed, the second one negates any conversion speed malus that occurs when the county faith has higher fervor than your faith.

Speaking of fervor: Have an eye on it and keep it high by avoiding religious wars, building or upgrading church holdings and by using the "council of Toledo" decision as often as possible. Low fervor not only slows down conversion (unless you have the icon of faith trait), but also increases the risk of your vassalls randomly converting to other faiths or heresies.

Converting each county on your own one at a time can be a tedious waiting game, but don't forget that your vassals also have court chaplains! Don't keep all those different faith counties yourself, give them away to vassals of your faith, preferably zealous ones, they will do the work for you while you work on your own domain first (vassals that are neither zealous nor cynical also do it quite often). I think high fervor also increases the likelyhood of your same faith vassals using the convert county councillor tasks on their domains.

Unfortunately, your faith being pluralist lowers both the conversion speed and the likelyhood of your vassals using the convert county councillor task on their own domains. If only there was a way to make Mozarabism not pluralist .... oh wait, there is! See 3.3

3.3) The special decision about Mozarabic reformation

If you are Mozarab and have at least 12 Mozarabic counties in your realm, you will notice two unique decision options popping up in your decision list to decide the future course of Mozarabism. "Bind the faith to Rome" (which also requires you to have the devotion level of at least "devoted servant" and have either a strong hook on the Pope or an opinion of 80+ with him) or "Break with Rome" (which also requires a devotion level of at least "Paragon of Virtue" and either a negative opinion with the Pope or sinful traits).

Both of those decisions will give you three options of how to slightly change Mozarabism at this occasion without it counting as a new seperate faith. Those decisions are final, you can't change Mozarabism again without creating a new faith once you have taken one of those decisions.

"Bind the faith to Rome" solidifies the Pope's position as your head of faith and offers the following options:
a) become righteous instead of pluralist and get the communal identity tenet
b) become fundamentalist instead of pluralist and get the pursuit of power tenet
c) stay pluralist, get religious law tenet and change the clergy appointment doctrine to spiritual/revokable

"Break with Rome" makes Mozarabism gain an own head of faith, but also lose the Ecumenism doctrine and offers the following options for change:
a) become righteous and get the pacifism tenet
b) become fundamentalist and get the armed pilgrimages tenet
c) stay pluralist, get the Islamic syncretism tenet and change the clergy appointment doctrine to temporal/revokable

I would not recommend to break with Rome if you want the achievement quickly and easily, because this causes a very high risk for Galicia with the holy site of Santiago (or the united Northern Spanish Kingdom, if you took the "unite the Spanish thrones" decision earlier) to become the target of a Crusade, especially if you still have a lot of Catholic counties in your realm that the Pope (whom you probably just made angry at you to get the decision requirement) would want to save from your heretic ways. I did try it in my playtrhough for roleplay reasons, because separating from Rome and becoming pacifist fitted the theme of the Detente ending of the Iberian struggle so well, and I became the target of a crusade almost immediately.

The best tactical option to get the achievement more quickly is chosing to bind the faith to Rome (which still lets you keep fairly good relations with Catholics and prevents you from being targetted by a grand crusade) and to go for the fundamentalist option. This greatly improves your conversion speed. Keep in mind that it will also make different faith rebell factions much more likely, so you should only do it if you already have a solid basis of Mozarabic counties and vassals so you can deal with a civil war. Also make sure to have high crown authority, otherwise your vassals will be at each other's throats like tribal barbarians, since the pursuit of power tenet grants the conquest casus belli.

Having a revokable court chaplain position is also tempting, but it isn't worth the conversion speed malus of staying pluralist, better go fundamentalist and revoke the position of inept bishops with a dagger if necessary. Going for righteous instead of fundamentalist would normally be a good middle ground to avoid rebellions while still having good conversion speed, but we definitely don't want the "communal identity" tenet for this achievements, which slows down conversion speed greatly in counties that aren't of our culture.