Upload the original song you've chosen into your digital music creation/editing software.
Isolate the notes of the melody, the bass line (if applicable) or the rhythm of the song that you'd like to modify for your remix.
Create a musical addition to complement the original song. When doing so, make absolutely sure the melody/beat matches the original song to avoid clashes in tempo/key, unless you're artistically inclined.
Splice the new musical addition to the song into the original composition. A typical tactic is to isolate what is being replaced in the original song and attempt to edit that section out, replacing it with yours. A new drumbeat or more pronounced bass line, for example, will sound more natural if the old beat/bass line is no longer present in the final mix.
Complete your remix and save it to a popular format, such as MP3. It can now be loaded onto MP3 players as well as most file-sharing programs on the Internet.