Note: This solution assumes you have already earned Ice Explorer by collecting your first Ice Diamond.
To earn this achievement, you will need to collect a total of ten Ice Diamonds in a single game. This will typically be the second achievement you earn. In the Icy Land (the blue area you start the game in), walk around to explore and collect any additional Ice Diamonds (the small, white crystals) that you see.
Soon you will also find monsters in the Icy Land that chase after you. These come in two types: the blue Yeti, and the orange Icewolf. To kill these, simply try to move onto its space during your turn. (Note that you cannot move NEXT to it, as then it would be able to kill you on its own turn! To avoid this, click on yourself to 'wait' and allow it to move next to you so that you can kill it).
Simply keep searching the Icy Land, killing any monsters in the way, until you have collected ten Ice Diamonds. Once you do, this achievement will pop. Remember not to leave the Icy Land if you are hunting for this achievement - remain in the blue area, and avoid travelling into differently-coloured areas. (If the music changes, you know you have left the Icy Land)
Note: If you are playing on Orb Strategy mode, you will instead need 25 Ice Diamonds. This is not recommended until you have a better understanding of the game, by which point you will likely already have earned this achievement.