Pickups can make a big difference. I have burstbuckers in 2 of my guitars, and although I think they sound great (nice clarity and separation even with the crunch cranked up) , they're not per se high output pups...particularly on the bridge. A Gibson 498T has higher output.
But if you want the honest truth (and I am saying this 'cause to a degree I'm in the same boat) I don't know if that Slash tone is possible with that Line 6 amp insofar as the amp distortion goes. Clean? Sure, they'll do a nice impression of a Marshall but that crunch is just not going to get it.
I have a Line6 Spider II 210 that I want to play out with and I'm looking into a pedal which offers tube driven distortion. Those go for typically between 100 to 300 bucks and might get you closer to the Slash tone right off the bat before new pups will. (But, they certainly don't hurt!)
Also...found this link to turkey talk about Slash's gear. Good luck...
_.-+~* Velvet Revolver *~+-._