What's involved in upgrading a circuit?
"Upgrading" the circuit involves running a new cable (of thicker gauge).  Basically you would be replacing a 15 amp circuit with a 20 amp, for a benefit of 5 amps but with all of the work of just running a new circuit (either 15 or 20) which would give you a total of 30-35 amps instead of 20.  Because you also have two circuits, if one blows, only loose whatever was on that circuit instead of everything.

Running a new circuit is the way to go.  As a commenter said, you can "balance" the load. Ideally you would not have the AC on the same circuit as your computer as the motors can put off electrical noise.

The gauge of wire you use will depend on the amperage and distance.  If your room is upstairs with the bedrooms, code will often require an arc-fault breaker to be used.