Tumblestone

Marathon is a part of the Arcade Modes DLC, legitimate runs at this achievement take about 15-20 minutes. The way the game mode works is you have to try to survive. If you make a mistake, the "playable" area moves down a row and will only move up if you clean out the play area. Once you clean out the whole field (minus the last row), you "emptied the board" getting that achievement and also get a lot of free points with wild cards.

Here is a video of me obtaining the achievement:


As for tips, I think that treating it like a solvable -- no mistakes -- puzzle (like in story mode) is the way to go. I've had two runs where I made a few mistakes early and I was playing for my life up until I got to around 1000 and that is not the way to go. You'll notice on this run I emptied my board multiple times early on and it gave me lots of free points. It also makes the game easier to work with. Toward the end, you do end up playing for your life usually. So once you make some mistakes and the gray area ends up going further and further down you have to try to play smart and make the "least costly mistakes".