well... you should probably check some videos and forum posts where people have detailed their builds first. there's at least a dozen ways to do it.
given what you have there, you would have been better off also getting the pro gpio assist board from pocketadventures, then it's essentially a matter of lining it up in the case and connecting "a" to "a", "r" to "r", "start" to "start", etc. once you have the pi soldered in.
you don't need that adafruit powerboost - the pro power assist has boost+charge built in.
if you are going to use that amp, you probably also want a low-pass filter to clean it up (pocketadventures also has one of those), but if you got the gpio assist pro, that has usb audio built in, which sounds much better. otherwise, the principal is the same as wermy did in his original video.
if you don't do the gpio assist, then you will need to solve the button mapping yourself (not too hard, but you'll need to understand the gpio inputs somewhat to hook it to the correct ones). that is like, for instance, wiring the "A" on the button board to gpio pin "5", etc. there are tons of resources on how to do that (for instance, search for "gpio controls" or "retrogame" from adafruit).
for video, those pins line up with that video board, once you've put the pi in the correct spot (on the front half of the shell). the hardest part is getting the flat cable to sit in its connector correctly, which is what you want to use a little bit of masking tape to do, to make the cable a little thicker and snug-fit.
conceptually, as-is, this is a lot like my second game boy color build, so the notes in there might be able to help you out:
viewtopic.php?f=43&t=8233
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