I suggest using an array and updating the position of the enemy(ies) on the array mathematiccally, and each turn or each tick, draw the in corresponding space in layout.
You could also use pathfinding behaviour for the enemy, and correct it's position each time it moves to create the grid feeling. You may also need to stop for a half second every time it moves some distance to create the grid-movement effect. Even though the pathfinding behaviour uses an unseen grid, I think it still is very smooth so movement will not look like grid based.
Just on the top of my head though.