I have a bedroom outlet that is controlled by a wall switch. I would like to make one of the outlet plugs "hot" all the time and the other plug still controlled by the switch. I have checked the wiring at the outlet and there are 3 wires. The back and white are attached to the two screw at the top of the receptacle and the bare is attached at the ground screw. Can this be done?

To maintain functionality by code you cannot. You can however leave it as it is, make the outlet hot all the time, or add another wire to accomplish what you desire.

To make the outlet hot all the time, you need to turn power off, remove the 2 black wires at switch then join them with a wirenut. Eliminate switch and install a solid wall plate cover over the electrical box.