Ape Out

Since this game suffers from crashing after the score screen and therefore erasing all your level progress and the banana you just collected i highly recommend you play this game in Laptop mode, you can select that in the apeout options.

The basics are: Collect the Banana (a little yellow banana spawns randomly somewhere one the map).
The ape picks it up by default when you run over it, it will be displayed in the apes hand as hes holding it.
You have to Finish the Level in one go from start to finish, picking up the banana in between.
I cannot guarantee the banana always spaws on normal levels but it seemed to always spawn on my arcade runs, which are harder but required for other achievements so you might try to stack them.
Higher chances if you currently still have the "no damage recieved" and "pacifist" badge in the top right corner in the pause menu.
It seemed like on the normal levels you have to activly search for the banana while on arcade they put it right on your path as a reward.

there are 5x2 bananas to collect.
5 on the normal difficulty (1 per level)
5 on hard difficulty (1 per level)
you can also get them on the equal arcade levels but its harder (but bananas seem to spawn more often)
You can check if you got the banana on a level if you look at the records and see a banana sticker on them.

General tips from my experience are hold the people with guns hostage and use them to kill other enemies. Try not to push your luck and rush. The bananas were in pretty obvious spots and didnt require roaming every corner.

Once you collect your final banana and finish the level it will start a minigame where you have to throw a barrel at the guy running up to you trying to steal your bananas.
If he gets to the top and takes them all the stickers will disappear and with that your whole banana achievment progress. I beat it simply by smashing all the buttons constantly because the game doesnt tell you how you even throw the barrel so i cant even tell you what button to press but just spamming should do the trick and you have unlimited barrels to throw.
If you fail the minigame you will not get the achievement. I havent tested if force quitting the app will let you keep your progress if you close it fast enough if you fail.

The current guides here don't contain all of the information you need to complete this. I haven't gotten this yet, but I'm writing this as future documentation for both myself and others.

You need to get a banana badge from all 10 levels (the 4 albums and break in, on normal and hard), and then complete a donkey kong minigame which appears immediately after collecting all ten. The minigame involves throwing barrels in order to stop mario from reaching the top - if you fail, you will need to collect ALL BANANAS again.

You can get a banana by reaching the B-Side of an album without dying (I believe this is also after you collect the baby in Break In). Once you do, the banana will spawn somewhere from level 5-8.

You can pick up the banana by grabbing it, but you can still grab and throw other things while holding the banana. The only way to lose the banana is by dying.

You may as well try to go for the arcade runs before this, just in case you can find a banana during those runs (since you will automatically fulfil the banana spawn criteria in arcade runs). If you don't find them during your arcade runs, you may as well use story mode to remove the time limit, allowing you to methodically search each level.

The banana looks like a small yellow object on the floor. It's not very big (much like an IRL banana) so keep your eyes pealed (no pun intended XD). As the guide above states the banana is usually in an area that you will travel through BUT IT IS NOT GAURENTEED EVERY RUN. It also ONLY spawns in the side B of every album.

Some numbers:

I got the banana first go of Subject 4.
I had to run High rise 22 times to see the banana and I NEVER saw it spawn in the first half, only the second.
Fugure took me 22 times.

That's as far as I've gotten.

Cheers