I know that the YouTube API has the capability to check if a certain user is subscribed to you. I was looking at doing something similar recently for a competition entry (You must subscribe to enter)
I don't know if anything exists already for what you want though. Also I worry it could be against some hidden TOS somewhere for making your users subscribe. I seem to remember reading that asking people to click like is against the TOS as you're incentivising likes etc... I'd do a bit of checking up in the YT TOS first before going any further.
Just a thought
Dave