In this sense it's fine - you're saying you want to catch up with someone (i.e. you're behind them and want to reach their position).
You could also say catch up to you and that might actually be a better choice, because catch up with you can also mean "have a conversation with you where we talk about all the things we haven't discussed yet", like when you haven't seen someone for a while, and there are things you want to catch up on. Catch up with you on Friends could be understood as "talk to you on the subject of Friends", which isn't really what you want to say!
Personally I think catch up to you with Friends feels the most natural and clear, and with just generally works. On can sometimes sound a little strange and might need a verb, like catch up to you on watching Friends. It's very subtle and I'm having trouble pinning down exactly what I don't like about on to be honest, it's definitely the right word in some ways! With is definitely safer.
edit - I know, it's because catch up on Friends is what you'd say when there's something you want to make progress with, where you want to get up to date with the show. But adding with you or to you makes it sound a little awkward, for some reason. English!