There are 5 different challenges in Dwarfs in which you must get at least 2 axes on:
[Included my video of 3 axes on all for reference]
1) To Battle! - In this challenge you have to defeat all enemies. A good strategy for this is to send two dwarfs into one of the weaker groups of goblins. You should have a level 3 dwarf after killing them. Send him alone to the next weakest goblins, and keep going around with just that one dwarf and he should reach level 12 by the end for 3 axes. The goblins go from green->green armor->red->red armor->boss
2) Catastrophe - For this challenge you must survive the lava/water onslaught for 3 minutes. Using the slow down ability seems to just extend the timer, so no need to use it. If you want three axes you must do so without using any dynamite. To accomplish this, it is vital to use the water to your advantage and seal it off in such a way that it blocks the lava streams. I did not do this so well in my video as I nearly lost, but hopefully the advice helps.
3) To the Point - For this you are required to reach a score of 125k (for the 3 axes) in 3 minutes. Speed up as often as possible and get ready to instant seal as dwarfs approach caves. If a shaman spawns and you can afford it, it is best to build an outpost nearby, make 4 warriors and shoot them at it. Shamans give fairly decent quick points so it's best to take them out quickly.
4) The More The... - This challenge requires you to simply reach 100 dwarfs really quickly. Under 4 minutes to get 3 axes. Simply build outposts near your main base and set the tavern on right away, and watch out for when they near undiscovered caves.
5) Godspeed - Well, this one wants you to survive for 5 minutes on 500% speed for the 3 axes. It also requires a score of 100000 but it is very difficult to get less than that if you survive. Do not worry much about score or build taverns. Watch your dwarfs and be ready to slow down time and stop lava/water ASAP. As soon as it is discovered, wall it off and send a dwarf to bomb the entrance. After that block it off when you have enough money, but if dwarfs near it, it's useful to block off that portion and keep an eye on it, or bomb the nearby dwarf, or simply send him away a bit.
Video: