What I like about the drums in rap music is that there is such a big variety. Everything ranging from live drums to 808 drum machines, to drum sample libraries are fair game when it comes to choosing where you are going to get your drum sounds. Often it is best to combine a variety of different sound sources. For example if you want your drums to have a deep kick and really boom than you should try to combine the boom of an 808 kick drum with a live kick drum on top. You just have to hollow out the low frequencies of the kick drum to make room for the 808-kick drum to fit in the mix. This will give you a clean deep kick sound that has some live elements to it. Perfect for Rap Music.
These same ideas can be applied to the snare drum as well. The characteristics of a live snare and a drum machine created snare are pretty vast. Generally if you want clean go synthesized, if you want grimy go with live. Beyond that there is a lot of gray area in between. Try combining the two. One of my favorite things to do is add a small lives snare to my 808 snares then send them both to the same bus channel and compress them together. This will glue them together and make it sound like 1 snare instead of two layered snares. The same idea holds true when you put reverb on your snare. Put it on both to help glue them together.
The other driving element in a rap beat is the bass. The bass line is can either be comprised of a live bass, or a synthesizer like a moog synthesizer. Stylistically speaking the different subgenres of rap beats tend to have a certain bass sound that is commonly used. For example, a group like "The Roots" is a live band and use's a live bass guitar in their tracks. Generally speaking the underground community of rap beat producers tends to gravitate toward the sound of the live bass. If you are doing southern rap beats or "trap beats" you are probably going to use a sine bass sound. This gives you a bass tone that is clean, deep and really grabs the subwoofer.
Typically the frequencies of an 808 kick drum and a sine bass will overlap. This in turn will create a problem for your mix when the kick drum and bass play at the same time. A common solution that engineers use to solve this problem is taking a compressor and side chaining the kick to the bass. What this does is every time the kick enters the mix it compresses the peak of the bass tone allowing both instruments to seamlessly fit in the mix.
If you want to take this side chaining trick one step further you can use what is called a "Linear Phase Compressor" which allows you to compress individual frequencies. What this allows you to do is compress only the frequencies that the two instruments share. This way you don't take away the entire sound of the bass, just the frequencies that overlap with kick drum.
Start applying these tricks and you will notice a big difference in your rap beats .