It requires rushing, so the help of a friend and a bit of preparation.
Your friend rushes you through all of the game. All you have to do is make a Town Portal at the relevant "quest object", so Andariel kill in Act I (if you are rushing multiple people, only 1 person has to come through and the others can stay in town, as long as you are all in the same party. Typically, I bring along 3 more characters, since I have 8 CD-Keys, but you can only have 4 connections to the game from one IP. With a proxy, you can fill the game up and then simply use all the other mules for Hellforge in Hell difficulty for a shot at a decent rune). Staff pieces in Act II, Summoner and Duriel. In act III you need to kill the Council and then Mephisto (no need to collect all those rubbish body parts, except for just the Flail), Act IV simply Diablo, Act V Baal.

Here it gets tough, your character will be ~5 or whatever, the ones sitting in town will be 1, but you need level hmmmm 25 I think to be able to speak to the Ancients. You thus need to find some noobie who is level 25+ and has not yet killed the Ancients and has not killed Baal yet. You persuade him to join your party so that your chars will get the quest (since as I mentioned earlier, all they need to do is sit in town in your party while someone does thei Baal quest) and in exchange you take him along for the run in Nightmare. The downside is that this person cannot see normal games any more, only Nightmare, though I think this was changed in the latest patch, so it should be better now? However typically people do Baalruns to 40 in Normal, then do their Ancients in Nightmare and they can go on to Nightmare Baals straight, until 60, when they move to Hell Baals. There can be quite a few people looking but sometimes you have to look for this person for an hour or so. You then repeat all of this for Nightmare and get someone else for BaalQ.

In Hell you get to Chaos Sanctuary (Act IV Diablo) and start doing CS runs. Open a game, join to it and have a bunch of others join. You go to River of Flame and kill all the way through to DIablo with everyone running behind you. Due to the exp algorithm in the game, you get an insane amount of exp, i.e. you can go from lvl 1-15 in one run. The downside of this is that at level 25, this gets very, very slow, i.e. you need about 8 or so runs to get from lvl 25-30. Either you can keep on doing this, or you can do Normal Cows, of which you only need about 2-3 runs max.

Once you are 30, CS experience picks up again at an insane rate and you can keep on doing those runs until... well whenever, but it's faster to get to 60 and then do Ancients and start doing Hell Baals until you can equip all of your gear (Enigma's STR/lvl is key here), and then do... whatever!

This was the standard way of doing it. There's also been a "new" method around for a couple of years now I think which involves Uber Trist. A high level char goes to Uber Trist and kills DIablo and Baal. Uber Mephisto spawns these annoying skeletal mages, all you need to do is trap Mephisto in this specific house so all he does is spawn minions full stop. You then have a Hammerdin there simply killing them all. Meanwhile the low lvl chars will be sitting in a corner and accumulating a ton of exp. This method is much better because you don't need to be making new games or anything like that and exp is ridiculously fast. I think you can go from 1-87 in 1 hour (but obviously you must have been Hell rushed first). The other advantage is that also high level players still get a lot of exp from this and it is not uncommon for people to sit in the game who are level 90+.

The downside is that you need an extremely well geared Hammerdin to do it. These runs can also be quite expensive, typically spots in them are sold on d2jsp for forum gold, but once you have an awesome character you can start selling them yourself.

So that's it really. If you're unlucky with finding Baal quests this can take longer, but if everyone goes swiftly and you have pro friends helping you, you can easily do 1-85 or so in 3-4 hours. Maybe try it out some time!