Choose the rap songs you want to mix. It doesn't matter if the songs are yours or someone else's (as long as you have permission). You'll need the individual tracks to the songs. Preferably, they would be the final takes. You can mix a finished song, but there are less options open to mixing that way.
Check that your individual tracks are labeled. At times, you may be dealing with a large number of tracks at once, so it helps to know what is what. What you will especially want to be looking for is the drums and the vocals.
Start by mixing the basics. It's no secret that rap music lives and bleeds through a driving rhythm. It's been like that since the genre was created. That means special attention needs to be paid to the two main emphasis points of the beat: the kick and snare hits. If those are sounding good, you are well on your way to rap gold.
Blend the other layers of music into the beat. The beat is the pillar that all the other instrumentation is piled up against. As rap music has evolved since it's beginnings, it's blended in new and more complex elements from different genres of music, just as other genres have borrowed elements of rap music. In today's rap song, it's not uncommon to hear full orchestration, ethnic instruments and beautifully complex harmonies.
Bring the vocals into the mix. Now that the band's all here, it's time for the star of the show to arrive. The vocals are going to be louder than everything. Don't go overboard with it. The music is still important. But it's also important not to let the lyrics get drowned out in the very heavy low end.