How to do all 4 speed running achievements?

Here's a good starting checklist:

1. Be at least level 45 or so for the normal mode runs. This will make sure you 1-hit almost anything, as well as have access to decent +movement speed boots
2. Do it with a group of dedicated friends. This allows you to all spread out and search for dungeons, ballistas, exits, et cetera. 4 people searching a map is always a lot faster.
3. Don't pick up loot. Rares and legendaries may be okay, but anything less just won't be worth your time.
4. Except for the rare middle-of-the-map objectives like the Khazra Dungeon, most objectives are on the edge of the maps. With 4 people, have 2 immediately take a left and a right while the other 2 beeline for the other side of the map and do the same. This should cut your time down by a significant amount.
5. This may seem obvious, but make sure your group uses the bathroom, smokes, eats, et cetera before you start. Bio breaks can kill your time.

Full disclosure: I haven't yet attempted these achievements, but I've put some thought into preparing for them.

• Bring a movement speed build. Things like Teleport on a Wizard, or Vault on a Demon Hunter with the +60% movement speed passive. Items aren't necessary, but they will help. If you're in Inferno gear with a decent quality weapon (600+ DPS), you'll be 1-3 shotting everything you encounter, so you don't need to worry much about variety.
• In that vein, bring large AOEs and piercing attacks. You want to be killing as much as possible with the least amount of effort.
• A group will absolutely help - even with 4 people in Normal, you'll still be 1-3 shotting everything, and even something as simple as someone sitting in town to accept/finish quests can reduce time. Being able to split up to find dungeons and for quests where you need to do multiple dungeons or one dungeon of a set (the Cemetery and Drowned Temple in Act I, the two Zulton Kulle dungeons in Act II, etc) is also a huge boon, and will cut down on your error margin significantly.

I've got all the speed achievements solo with my Barbarian, my advice is mostly Hyppys' with a few minor differences.

1) Levels are good but spec out for movement boosts as well, Barbarians leap is excellent for crossing gaps.

2) Avoid combat if at all possible, the less time you spend hitting things the better. Thorn type armour is good for this, or run a monster train behind you - remember they're as fast as you. If you have a fire and forget AoE (Earthquake in my case) dump that every now and then to thin the pack.

3) Focus on the speed run, not the loot or the random dungeons\spawns. Don't forget there are a few Bonus quests that you can happily ignore

For team runs use the games movement capabilities wisely, Town Portal, Waypoints and Banner teleports. For example, while three people are running through Khazra Dungeon have the fourth stand at the exit path. When Khazra is finished it's quicker to Town Portal back and use the banner to TP to player four.