Apart from completing all content, there's currently one achievement for doing a challenge run:

Speedy: Reach the tournament after playing only 20 matches or less. 

This counts every round started, whether it was won, lost, tied, or abandoned by turning off the game.

However, Puzzle Battles don't count. Play those all you want.

You can post here about your attempts at this challenge, or ideas for other challenges. I hope to add a couple more achievements eventually.