First, you will need to find who the original copyright owner of the song is. For example, one artist's song may have multiple songwriters, and you will need permission from all of the rights holders on any track or piece of music. That means all their publishers, and/or the record label that originally released the music.
Once you have found out who those rights holders are, you can either try and contact them directly. The best way is to find the artist or songwriter's music publisher or collection society and to contact them enquiring about permission to use the music.
It’s much more straightforward to release a cover version, provided you don’t change the lyrics or melody, if that’s a route you’d like to try, read our article on what you need to release a cover version.