Tom's actually a got great voice - most of the stuff from Swordfishtrombones and subsequent albums are just heavily distorted. As mentioned, his early stuff is much softer - it's basically jazz/trubadour, with the odd beat poetry track thrown in. Of course, even today it really isn't any more gruff than Satchmo's was. Sure, it's not all smooth and processed sounding, and he's far from being "diva" material, but it's got a hell of a lot of character, and is absolutely perfect for the stuff he writes - I'd direct you to 'Come on up to the house', 'Kentucky Avenue' and 'Ruby's Arms'. It's absolutely not for the kind of person who's into mass music, though, since it's not really designed to be as accessible as your average rock album.
His voice changed after several years of whiskey, and boozing his way around Europe. Closing Time was a fairly big smash, but eventually he was no longer the darling of the music biz and his label treated him poorly, so Tom sort of went "anti-pop". The stuff since Swordfishtrombones has been about as un-commercial as it gets, really, particularly Bone Machine and The Black Rider.
So why's he 'popular'? Because he DOES his own thing, and does it well, even though it isn't really all that commercially viable. He's a musician's musician (look him up at AMG and check out the lists of artists that have covered him - the Eagles, Springsteen and Rod Stewart all had huge hits with Waits' songs), and the kind of artist critics dig - just listen to Swordfishtrombones and realize it came out in
1983. It was contemporary with Duran Duran and Flock of Seagulls! You'll also hear his stuff VERY regularly, as background music in non-Hollywood film. He also keeps reinventing himself, very successfully, something that isn't easy to do. And probably most importantly, he ranks right up there with Dylan and Cohen as great lyricists. His songs embody what he's writing about - like you said, he sounds like a guy living in an alley making music on trash cans. That can also make his work sort of unaccessable to most people, though. Especially first time listeners.
For better albums to start out with... Closing Time is damn good, and will give you an idea of his style from '72 - '82. The Asylum Years would also be a good choice, since it's sort of a Best Of from that period. For the later stuff, Beautiful Maladies is good in that it includes tracks from his albums from '83 - '93, but it isn't very cohesive as an album. Still, it's a decent overview of the later stuff, so if you hate it, there's not much need to get the albums. Mule Variations is a pretty good introduction to his later stuff, too. It's got a lot of the harshness of Bone Machine and Black Rider, but there's a couple of fairly "rock-ey" tracks, plus a few softer, more emotional ballads. He's also got a pair of albums coming out on April 9th (hoorah!). From what I've heard about them, you'll want to try something else first.
Oh, and speaking of Waits' bum character, for the other Waits fans here, pick up Gavin Bryars' "Jesus' Blood Never Failed Me" - it's an evolving instrumental piece with Waits "duetting" with looped snippet of a homeless man that was recorded for a BBC documentary. Very, very good, though it's
really not for everybody.