Is it possible to boost an electrical circuit for more current draw?
To directly answer your question: no you cannot "boost" a circuit. 

Every wire has a voltage drop, and according to NEC it can't be more than 5% at the outlet. Voltage drop is proportional to the length of the wire, resistance (which is dependent on the size of the wire), and current. In other words, as your load goes up, so does your voltage drop.

If your existing wiring is just on the edge of either load or size, then the current may be going up enough that it's below the UPS threshold and thus the UPS is switching to back-up power because it sees it as a brown-out. 

If the cable is too small (typical is 14AWG for a 15A circuit, 12AWG for 20A) then the only thing you can do is either increase the cable size, or run a new circuit (and really, you're not going to pull out the old cable and run bigger cable, it's just as much work to run a new circuit so you might as well do that). 

If the load is too high (eg, you're close to 15A) then the only thing you can do is run a new circuit, or decrease the load. 



You could start by using a Kill-a-watt or similar device to measure power consumed by your PC,  printer, TV, etc. This will help guide you to what is consuming too much power, and where (if you go this route) it makes sense to split the circuit up. 

You mentioned lights; using LED or CFL bulbs may be a stop-gap to save some power, and is certainly one of the easier things you can do.

If it comes to it, there's ways to be creative about how to split the circuit up, without necessarily having to rip out a lot of drywall. You may be able to run a new cable halfway, and then splice in and split the circuit into two, starting from a light or receptacle. You may be able to find the wire underneath in the basement, and cut it and put in a junction box so the circuit from that point on is on the new circuit. It's hard to be specific here without actually being in your house.