Options for Running High End Computer on Old Circuit
TLDR: Pay your landlord to add a 20A dedicated circuit for this PC. 

Fuggedaboutit. 1350W is just too much.

Remember, you're talking 1350 watts at the low voltage DC side - you have conversion losses. You must provide what the AC-side requires. That will be clearly stated in the specs, or placarded on the unit.  Here's a Rosewill Photon 1350. 



Let's be gracious and assume this mark is saying 15 through 8 amps, not 15.8 amps.  That is the actual power it will be drawing. 15 amps = 1800 watts, which reflects the internal inefficiency of the supply, and that's about right.   This is a continuous load, so you must provision 125% of that power, so that's 18.75 amps.  This will take the entire output of a 20A circuit.  

You don't have one of those. 

If your "12 amp" figure is based on a 15A circuit you are applying the same 125% derating to, then good on you for working it backward like that.  Regardless, a 15A circuit cannot power this.  Running a circuit at 100% load continuously will result in breaker trips, which results in death. Permanent death if you're playing Diablo hardcore. 

Beware old service panels

If your 80 year old building has a Federal Pacific, Zinsco or Challenger panel, then maxing out a circuit is madness. Those breakers are not reliable.  This could result in fire, and perma-death of you. 

A Challenger panel can have its breakers changed out to Eaton BR/C, (C for Challenger), and it'll be fine. Zinsco and FPE are beyond recovery, as not only are the Connecticut Electric breakers worse than the originals, but the buses themselves are defective. 

Landlord permission + licensed electrician is mandatory

You cannot alter the wiring in your rental unit.  I live in one and the most I've ever done is change a cover plate.  All work requires landlord approval because of a legal concept called "mine / yours".  And all work in a rental unit must be done by a licensed electrician, to disallow low-quality work by cheap slumlords or sloppy tenants.  So if you want to upgrade the electrical service to the unit, you'll need to have a sit-down with the landlord and pay a 3-digit sum for the electrical work.  Or move. 

Another snag is that older buildings are often laughably underpowered.  I have seen units with 30A/120V for the entire unit.  You will not be able to add a 20A circuit to such a unit.  

Further, if the landlord pays for your electricity, expect a rent increase that may not be fair, because the landlord does not want you bitcoin-mining.  

Don't forget air conditioning

Putting 1800 watts into a room will make a lot of heat.  5300 BTU/hr to be precise.  You will need an equivalent amount of A/C just to break even.  That will need its own circuit, since sharing the circuit with this PC is right out.