Slash has a natural tone but for the most part hes using lots of volume and gain on the overdriven channel of his marshall.
In the example of sweet child o mine, the only reason the intro sounds so clean, despite being on 'dirty' settings, is after each note is picked its muted to prevent it ringing out. With only one note being amplified at once your preventing distorition and break up.
Ive seen people barre the 12 fret for the intro and it sounds awful this way as the notes turn to mush!! Before you pick each note in the sequence you must lift your finger off the previous note and let it lie against the finger which just fretted it to mute it.
If its still sounding too dirty then turn the treble/gain/presence down... but really these are important components of his tone.