to me, having a good sample is the most important thing. there's a lot of shytty, cheap ass 808s out there. quality sound sources (of any instrument) are the foundation. "you can't polish a turd".
but as far as this goes, yeah, yeah, yeah saturation, eq and compress it, blah blah blah. sure, doing that can definitely help sculpt the sound, don't get me wrong. but lemme clue you in on something, the big reason it sounds 'so clean' in the mix is because there is space for it, all the other shyt is moved out of the way. that low end belongs to that 808.
and that means not putting all your attention into the 808 and trying to force it in the track...you gotta go to all the other instruments and start removing/eqing shyt that's competing with your 808's frequency range. when you open up that low-end for it, the 808 will breathe.