Actually easy to do with an interface. If the music (backing track) is on your computer already, you just need to insert the file into a track in your DAW, then record your voice over into a new track, then mix.
If the music (backing track) is in an outside device and you want to record it into your computer/DAW at the same time as your voice, this will require an interface with at 2 least 2 line inputs and one mic preamp that can be used simultaneously. Plenty of them around in the sub-$250 range.
If you don't need to record your voice at the same time as the music (backing track) you can get away with an interface that will allow 2 simultaneous tracks to be recorded, has 2 line inputs and 1 mic preamp (a shared input might work). There are many interfaces around in the sub-$200 range that will do this.
The mixer solution that ecc83 proposes would be used for an external source of the music plus mic, but woudl nto allow you to adjust the vocals after recording because they would be on the same (stereo) track as the music.