XCOM 2

Bonds are formed and strengthened based on the compatability of a pair of soldiers and the number of missions they go on together. View a soldier and choose "Soldier Bonds" to see which other soldiers he or she is most compatible with. Every soldier is unique - some have high compatibility pretty much across the board, and some are only compatible with that special someone. I don't know if this applies in all cases, but when I played with the tutorial enabled, my Jane Kelly had a high or very high compatibility with literally every other soldier in the barracks, so she may be a good one for you to single out for this achievement.

One more thing to keep in mind when matchmaking your soldiers is whether they are in the same tactical class. I, for example, pretty much never send two rangers out together, so I wouldn't want to bond a pair of rangers.

Bonds increase by sending compatible soldiers into combat missions and covert ops together. Certain events that occur in combat, such as rescuing an unconscious squadmate, or surviving a bloodbath where more than half the squad is killed, can give the bond a boost and even make soldiers more compatible. (Keep that in mind in case you have a pair of soldiers you really want to like each other, wink, wink.) Note that even soldiers with very low compatibility will eventually become bondable if you send them on enough missions together; it just takes longer.

OK, so you'll get a notification when soldiers are ready to bond. For the Level 1 bond, you just go to "Soldier Bonds" again and select the soldier's new bondmate. Simple.

To get to Level 2 and Level 3, you need to keep sending the soldiers on missions and/or covert ops together, and you also must have the Training Center built in the Avenger. Keep an eye out for the notification that your pair of soliders are ready to take their relationship to the next level, and then send them to the Training Center. This will make them unavailable for missions and covert ops for a few days. In order to use the Training Center, both soldiers must be on the Avenger (not on covert ops) and not be wounded. After they complete their Level 3 training, the achievement will unlock.

(Edited to include covert ops as per HirsuteDave's comment.)