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.