Select your songs. It's important that they are of a similar tempo, as dramatically altering one to fit the other would sound odd, and likely ruin the overall effect of your mashup. Try playing one of the tracks and humming or singing the melody from the other song at the same time. This will give you a good idea of whether the tracks will fit together. If you have to dramatically alter the pitch or tempo of the song you're singing in order to make it fit, you should probably choose another.
Launch the multitrack audio software and create a new document or project. Use the "Import audio" feature, usually located in the "File" menu, to import your audio files. There should be four in total, an acapella and an instrumental file for each of the tracks. Place each file on a separate audio channel. Mute both parts for one of the songs so you only hear one when you press play. Use the drag control to ensure that the vocal and backing track parts are correctly synchronized. Mute the first song and repeat the process with the second one.
Mute both of the vocal tracks and select one of the instrumentals. Right-click the audio file and create an instance of the time-stretching plug-in. Move the speed slider up and down until the drum parts of both tracks match each other. Once they are synchronized, make a note of the speed percentage applied by the plug-in, it will be a small positive or negative figure, for instance +8 percent or -5.5 percent." Unmute the vocal track, and create another instance of the time-stretching plug-in on the second track. Apply the same speed percentage processing to the vocals so all four tracks are now synchronized with each other.
Play all four tracks together. Chances are they will sound overly busy and chaotic, as there is far too much happening at once. Use the scissor tool from the edit menu to cut out individual lines and bars from each track, leaving only the parts you want to keep. You should be able to switch between the two backing tracks and the two vocal parts seamlessly, as they are all synchronized with one another. Play with the structure until you have an arrangement that combines the vocals from track one over the backing from track two and vice versa.