Normally you'd press the Action key to select it; on a computer this would be the X key or Enter key. The game wasn't designed for tablets so instead you'll have to click around at the bottom to select a character.
How the selection screen works is the player is actually invisible about 4 tiles from the bottom and selecting is just them walking right or left. You'll have to click on where the player is to make the selection. So just click around 4 tiles from the bottom until it works.