need to find a way to get a hold of the stems (track-outs). The stems will separate all of the instruments and vocals in the original song so you can rearrange and remove/edit them. The stems may be sold by the original artists, or found ...blackmarket style, somewhere online, but this way wont give you legal permission to sell the remix.
Or if the artists has a moment of only vocals without music behind it, you can grab that part, maybe? or maybe you just take the original song and add new drums to it, new melodies, etc...cut and rearrange the beat into something new. Ya just gotta be creative...