Double Dragon Neon

For this achievement you need to get all 20 songs (10 Sosetsitsu songs and 10 Stance songs) maxed out.

To be able to upgrade your songs, you need 'Mithril', which is what every boss you come across drops, I'll list how much Mithril is dropped for each difficulty.

Normal: 3 Mithril per boss.
Dragon: 7 Mithril per boss.
Double Dragon: 17 Mithril per boss.

And the songs have to be upgraded 4 times for it to be maxed out as well, to upgrade your songs, you have to do it at the 'Tapesmith', which is located on 2 different stages, the first one you'll come across it, is on the 2nd stage and then the second one on the 9th stage. Here's a list of how much Mithril you need to upgrade your song(s) each time.

First time: 2 pieces of Mithril.
Second time: 6 pieces of Mithril.
Third time: 10 pieces of Mithril.
Fourth time: 12 pieces of Mithril.

Overall total per song: 30 pieces of Mithril.

Overall you'll need 600 pieces of Mithril, which is a lot and if you're going to grind for it, which most people will want to, it's going to take you some hours, if you're going to do it in one sitting.

Anyway, in my opinion, the best way to do this is to keep repeating the 2nd stage on the Double Dragon difficulty, since the boss drops 17 pieces of Mithril each time. Also once you've killed the boss and then repeat the stage, you can stop in the Tapesmith to upgrade your songs, or you can wait until you get all the Mithril.

For the Mixtapes, I'm pretty sure they are random drops, these will probably take just as long to grind then the Mithril, since you'll need 50 Mixtapes per song. Also even if you've got 50 Mixtapes of a certain song, they will still drop from the enemies, which is quite annoying especially if you need to collect for example 1 more Mixtape for the achievement.

The best stage to farm money, seeing as it is helpful to buy the Mixtapes as well as just farming them from enemies, would have to possibly be Stage 9. I just recently went through both Dragon and Double Dragon difficulty and these were the results.
Dragon: $9,700
Double Dragon: $12,600
You could possibly get more then this, but I do recommened to do this stage, also making sure you open the chest and keep hitting the Tape Worm for Money and also Mixtapes.
I've just done a run on each difficulty (except for Normal) on Stage 10 and these were my results;
Dragon: $13,100
Double Dragon: $15,300 (Without the chest after Skullmageddon, since the bastard ended up killing me when I was so close to killing him).
Anyway, what I mean by the chest after Skullmageddon, if you don't know about it, is after you've defeated Skullmageddon the first time on Stage 10, you'll have to kill him a second time, which is in space and you become 'Ro-Bro'. As soon as you appear in space, go left and there should be some health there, but what you want to do is go all the way to the bottom left of the screen and push down, it'll take you down a ladder with 3 chest and 2 keys, which is some extra money, credit to Metamania2313 for sharing this information.

Shops also sell Mixtapes as well as 'Health' (Crystal Dragon Kick) and Extra Lifes, here is a list of what stage and what Mixtapes the shop sells on each of the following stages that have shops;

Stage 2:
This shop is located to the left when going up the first ladder you see, it'll be in screen range anyway.
[Sosetsitsu]Fireball ($900)
[Sosetsitsu]Spin Kick ($900)
[Sosetsitsu]Dragon Swarm ($1,100)
[Stances]Balance ($1,100)
[Stances]Training Wheels ($900)

Stage 3:
This shop is a little tricky and hard to find, it's near where Skullmageddon announces for his men to greet you on the bridge, but before you jump over the hole, you'll see a part where it's got yellow tape saying 'Do not enter!' and there is a vent below it, go up to the vent and hit it a few times and it reveals the shop.
[Sosetsitsu]Bro Dozer ($1,000)
[Sosetsitsu]Bomb Toss ($1,000)
[Stances]Magic Gambit ($1,200)
[Stances]Successive Strikes ($1,200)
[Stances]Desperation ($1,200)

Stage 5:
This shop is after the Killacopter, it's located to the left on the ladder when you go down it, along with a Sickle on the ground in front of the shop.
[Sosetsitsu]Spin Kick ($900)
[Sosetsitsu]One Inch Punch ($1,000)
[Sosetsitsu]Song Snatch ($900)
[Stances]Power Gambit ($1,200)
[Stances]Weapons Up ($1,100)

Stage 8:
There will eventually be a set of falling platforms that drop down when you walk on them, once you reach this part, stand on the third falling platform and let it take you down a bit, once so, jump off to your right and the shop is right in front of you.
[Sosetsitsu]Dragon Swarm ($1,100)
[Sosetsitsu]Bomb Toss ($1,000)
[Sosetsitsu]Healing Touch ($1,000)
[Stances]Rage! ($1,300)
[Stances]Desperation ($1,200)

Stage 10:
This shop is the same as Stage 8, there will be another set of falling platforms throughout the stage, do the same thing, stand on the third falling platform, let it take you done a bit, then jump off and again, the shop will be in front of you.
[Sosetsitsu]Knee Drop ($900)
[Sosetsitsu]Lightning ($1,100)
[Stances]Stunner ($1,200)
[Stances]Successive Strikes ($1,200)
[Stances]Absorb ($1,300)

Overall cost for x50 of each Mixtape;
(I know you won't be spending exactly this much on the Mixtapes because enemies drop them too of course, but I'm just giving the exact prices of how much you would be spending if you were to be spending this sort of money on Mixtapes).

[Sosetsitsu]
- Knee Drop: $45,000
- Bro Dozer: $50,000
- Spin Kick: $45,000
- Fireball: $45,000
- Dragon Swarm: $55,000
- One Inch Punch: $50,000
- Bomb Toss: $50,000
- Lightning: $55,000
- Healing Touch: $50,000
- Song Snatch: $45,000

[Stances]
- Balanced: $55,000
- Training Wheels: $45,000
- Power Gambit: $60,000
- Magic Gambit: $60,000
- Stunner: $60,000
- Rage: $65,000
- Weapons Up: $55,000
- Successive Strikes: $60,000
- Desperation: $60,000
- Absorb: $65,000


That's all, thanks for reading and good luck for the achievement and hopefully all this information helps.
Also if there is wrong information, please let me know so I can fix it up, I like to have my guides neat and tidy.