How to do all 4 speed running achievements?

Solution 1:

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.

Solution 2:

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.

Solution 3:

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.

Solution 4:

A lot has changed in the game since this question was asked: Reaper of Souls was released, difficulty systems overhauled, and Loot 2.0.

No doubt, the run will be in normal.

Correct: only attempt this on Normal difficulty, there is no achievement for clearing it quickly on higher difficulties. I would also add that to avoid wasting time on a failed run, it is best to attempt this only after the following:

• Character is level 70.
• You can clear a greater rift on the equivalent of Torment I or higher difficulty (level 10) well ahead of the 15:00 bonus timer without being in any real danger of dying.
• You have a few Paragon levels: certainly a minimum of 40, to get 5% increased movement speed.
• It may help to use Kanai's Cube for some defensive abilities to augment your gear and build. For example, Krelm's Buff Bracers offer a knockback and stun immunity, which might help shave a few seconds as well as annoyances off your run as you blow past elite packs. Browse through the cube's list and see what you have that might help.

It should not take a lot of time to meet these guidelines on a level 70 character if you do not already meet them. Run bounties for materials, and rifts/grifts for gear. This is pretty bog-standard "new level 70 priorities" that are easily searchable on sites like Reddit and Icy Veins. Even if you are starting fresh, a few hours of intense, targeted farming should get you in shape to speed-run this content.

The material I covered above is new since the question was asked and the existing answers provided. That alone should be sufficient to blow through the achievements well ahead of the hour time.

Should I invest in a movement speed build?

This is a bit subjective, but I have completed these achievements without a special build. I certainly have some movement abilities (e.g. Teleport with the Wormhole rune), but building for speed is likely not necessary as long as you can plow through monsters without taking noticeable damage (see my second point above).

Should I do it solo or have a group?

Other answers mentioned some great tips, including group strategies. In general, I find that as long as you are familiar with the layouts of the maps, have decent gear and movement speed (again, see my second point above), and do not waste time picking anything up that is not a legendary (orange or green), you can likely do it solo without the logistics of finding a group willing to devote the time for the achievements.