I recently registered the domain "iFamilyGuy.com". I'm currently developing it as a Family Guy episode website, where members can watch full length Family Guy episodes.

Now before I purchased the domain, I checked out many other competing websites and found that most of them contained "Family Guy" somewhere in the domain. It appears Fox (the owner of the TV show) has given leeway to many other developers.

Do you think I could run into legal problems monetizing the website?

Yes, i believe fox go to great lengths to shut down such websites... putting it simply its piracy, more than that you are the distributer of piracy. Personally i love these sites, keeps me entertained when i get bored but you better watch your back!