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.