I am driving a 4-digit cc-type multiplexed display. The chip, after touching the surface, appears to be hot. Is it normal?
The microcontroller should not be getting hot. How much current are you trying to get out of it's output pins, in total?
You should be able to calculate the expected power dissipation (roughly)... for each pin, take the current you're getting out of it, look that up on the graph in pin driver strength section of typical characteristics chapter of datasheet to get an estimate of the voltage drop between that pin and the supply rails under load (or measure it). Multiply current by voltage drop to get watts dissipated in the chip for that pin - add that up for all pins in use - what do you get?
Are you using series resistors or driving the display straight from the Mega pins? I used a 4-digit display and had each digit driven in turn, and used 100-ohm resistors. The chip (a SMD 328) got slightly warm. As long as the chip is not hot, you're fine.