I have this idea I’d love to do where a character has a small mimic as a pet, like the size of a clutch bag. I’d name them Purse. 

I thought I could give them small items to hold, maybe they could attack people who try to rob me, turn into different things, etc

maybe he grows in size the longer I have him and could turn into better things or do more damage?

I’ve never homebrewed something before.

How do I start making him? And what would be the best class to use him with?

No homebrew required you can use Warlock with Pact of the Chain with an imp familiar. Just say that the imp familiar looks like a mimic.

Alternatively, you could use the Strixhaven feats for the Quandrix specialty familiar.

Ooh, this is such a fun idea! Can I steal it from you?

Go ahead! Just let me know how you use it!