Well it's a bit of a minefield but you will at least require a MCPS (Mechanical Copyright Protection Society) license, and if you're going to screen the video containing the music publicly then you may also need a PPL license (Phonographic Performance License). I suspect in your case you will need to purchase the Limited Manufacture License from them.
Limited Manufacture Licence (LM)
Note that the above license doesn't cover music from operas, musicals, ballets etc. This requires an additional license again.