The idea of using Polymorph to get around the limitations on Awaken has been covered before. It has a bunch of issues, having to do with the fact that eventually you'll want to unpolymorph the thing back into its original state and (preferably) keep the int boost and the free language, and there's a lot of reasons that might not work the way you want. Find Familiar, though, gives you a critter that basically is a low-CR beast, other than the fact that it's not actually a beast, so there's much less incentive to change it back when you're done. You can't normally abuse True Polymorph to permanently upgrade your familiar in a meaningful way, because you can't increase CR with that spell… but changing it into its actually-a-beast version, or into some other beast of the same CR should still be doable, and that seems like it would make the creature a valid target for Awaken. At that point, it seems like you could just… leave it as it is. It would be a beast instead of a celestial/fey/fiend, but that seems like a small enough price to pay for cranking the int up to 10 and giving it the ability to speak a language. There are a number of other useful features of familiars, but those all seem to be facts about what the caster can do, rather than things that would be disrupted by True Polymorph.
Basically, this looks like cheese. It's a way to permanently upgrade your familiar with spells, and it seems pretty clear that the authors have made efforts to try to prevent exactly that. At the same time, as far as I can see, it's viable-within-the-rules cheese. Is there anything I'm missing here?
The argument "it's obviously cheesy so as DM I wouldn't allow it" is not a acceptable answer to this question unless you can also find some in-the-rules way to challenge its validity. If doing this would cause you to lose out on any of the standard features of a familiar, that would be useful additional information.
Best Answer
If you're got true polymorph, it seems to me that the ship has pretty much sailed on letting you permanently upgrade things with spells. You can transform the whole party into CR17 Adult Gold Dragons; you can turn pebbles into Evokers or Young Silver Dragons or Unicorns, and just keep doing that once per day forever. Giving your familiar a 10 INT seems like not something your DM should be worried about at that point in the game.
One drawback: one of the uses of your familiar is as a disposable scout: you can send it to check for traps, and if it dies, the only cost is you need an hour and 10gp to resummon it. (And it's a ritual spell, so you don't even lose a spell slot!) But if you're using this trick, then true polymorph falls off as soon as the familiar drops to 0 hit points, and that probably means that you lose awaken, and awaken costs 1000gp to recast.