Sorc is definitely the way to start out if you want to ever get yourself into the economy of high end items without paying real cash. Hammerdins are better at Baal runs, which is probably the fastest way to get high end items, but their effectiveness in Hell mode is highly dependent on gear that you will neither find nor be able to trade for by the time you reach level 70-80.
Cold sorcs can be a pain to level until you get frozen orb (thats the top tier cold sorc spell iirc?), but are definitely the least gear dependent MF character you can make, and the gear you need to turn them in to a top tier MFer is fairly cheap and you will likely find pieces of it as you level. My recommendation is to find someone to rush you through normal and do Baal runs until they don't give much exp (I don't remember what level that is), then do the same for nightmare and hell. Normal mode can be hard to find a rusher for if you don't have a friend, the normal pay for a rush is your Hellforge drops (gems and rune words), but normal mode drops are crappy so there isn't a lot of people out there bothering with it. Nightmare is a lot easier to find a rusher because the Hellforge drops are decent enough that theres an economy for them. However, hell mode Hellforge can drop even the most valuable rune words (again, iirc, my memory is a bit fuzzy), so you may not want to give them up, as a top tier rune word could pay for an entire top tier set of cold sorc MF gear and more.
Anyways, once you get to level 60-70 through Baal runs, you can start doing nightmare Mephisto/Baal runs on your own or rush other players for their Hellforge drops, gather up some currency to trade for top tier MF'ing gear. Once you've got a decked out MF'er, you can take on hell mode runs solo (though if you go pure cold, beware of cold immune mobs - some people go hybrid frozen orb / meteor, it seems to work just fine but I never bothered to make one), gather up even more currency and eventually find / trade for the items for a Hammerdin or whatever character you want to build.
Alternatively, you can just play through the game the way it was meant to be played and enjoy yourself

You might want to stay out of multiplayer if thats the case though. I've rebooted my D2 career a number of times in the last few years, and theres very little community around just running around and playing through the game normally as a group. Playing on multiplayer servers will get you spammed with bots and all sorts of crap, and you'll have random twinks run into your game with the absolute best gear they could possibly have at their level (purchased/found from their other high level chars of course), who are around your same level but just rip through the mobs like their made of paper. Or people join your game and go do your quest objectives while you aren't there, making you have to restart a new instance because you didn't get credit for it. Then of course theres the problem of almost everyone having auto-loot hacks that let them pick up every item of any value instantly before you even realize what it is. Oh, don't forget the constant spam, in chat channels, in game, and in whispers even when you lock your game out from anyone else joining. Honestly, the last time I tried to restart D2, thats literally the reason I quit and I hadn't even made it to level 10. Its just annoying as all fucking hell to have half your screen constantly covered in spam.