just be careful with that model you dont want to keep it where dust can get in the grill on the heatsink it could be costly as they are prom to overheating and causing the gpu to fail. take it to your local repair store to get maintained.
Test it with the battery out, try one stick of RAM at a time, then try and reset the processor , test it to an external monitor that's the testing i do. if they fail then you know you have a logic board problem.
try resetting the digitizer on the logic board ifixit has a guide on how to do it, if this does not fix your issue then you might want to consider a new display. it a may also be a board issue. happy repairing
connect it to itunes and do a factory restore on it, it may be software corruption, if not strip the device and use WD-40 contact cleaner if this doesnt solve it there may be a chance its the logic board
press and hold the home and on/off button until the apple logo appears then release your finger from the on/off button while keeping your finger on the home button until the connect to itunes screen appears. "it might take a few goes to get this right" happy repairing