Best suggestions here push towards the new Dedicated Circuit - and as mentioned, if you were to bond the Receptacle to the Equipment Ground at the panel with one #12 cu, you may as well bring two more #12 Conductors with it and have a nice clean, Dedicated Circuit to drive your PC and Peripherals from.
This Dedicated Circuit could also eliminate UPS cycling, due to Transient Voltage sags / dips caused from other loads connected to the same Branch Circuit (like when an Electric Motor starts, it causes a temporary drop in the intensity of Incandescent lamps).
The UPS will operate (and the PC also) without an Equipment grounding Conductor, but the benefits of noise rejection, sheilding and most of all - almost zero shock to ground hazard, will require the EGC connection.
Lastly, as mentioned by others, when you replaced the 2 wire Receptacles with three wire ones, you have setup a potential hazard for equipment with three wire cord caps (Plugs) which have metal cases (or some other method for leakage currents to flow through a person's body from an internal wiring fault).
Either replace the Receptacles with two wire ones, or protect the "String" of three wire Receptacles via a GFI Receptacle (or Circuit Breaker).
The GFI Receptacle method would require placement of the GFCI type Receptacle device at the first outlet of the Circuit (first outlet from the Panel). Place the 2 wire Circuit coming from the Panel on the "Line" side of the device. Place the outgoing 2 wire Circuit on the "Load" side of the device. Now the Receptacles which are "Downstream" from this outlet (along with the GFI outlet) have Ground Fault protection. Be sure to add the "No Equipment Ground" stickers, which come with the GFI device, to the protected Receptacles.
If you choose the GFCI Circuit Breaker option, just find the 2 wire Circuit for this string of Receptacles (the Black and White wires), and run them through the Breaker.
If the Circuit is a three wire / Multiwire Circuit, cannot use a 1 pole GFCI unit (need a 2 pole unit or resort to the GFI Receptacle method).
This information is being given to you as a reference to what options you have. Strongly suggest you have this work performed by / under the supervision of a Qualified Electrician!!!
Working in live Panels is VERY DANGEROUS!!! It only takes one slip-up, and before you even realize it, things have taken an ugly turn!!!
Incorrect connections of Circuitry will result in safety hazards plus damage to equipment.
Hire a good Electrician to do this work for you, and observe the installation being performed to see how its done.Be sure to mention to the Electrician you would like to observe the installations!!!
Nothing makes a Technician more nervous than having someone watching over them, as if they are "Supervising" the installation. Let the person know that you are interested in observing the correct install methods, and chances are the person will not only feel comfortable with you observing, but go as far as describe the items and steps involved.
Works out for both of you