The "1300w" power supplies for PCs are apparently a specialty item made by a few manufacturers. That nominal wattage is the power it can give to PC components; due to inefficiency, it draws more from the grid.
A PC doing heavy computation is what's called a continuous load. For those, you must allow an extra 25%. For instance if the load is 12 amps (1440 watts), then 125% of that is 15A. Now, 15A is the common circuit size found in houses, and power supply manufacturers know that. The power supplies are designed to draw 12 amps because of this rule.
The ones I studied are switching power supplies (required so they can hit the "Gold" power factor), and they can also run on 240V. At twice the voltage they take half the current, or 6A. Two of those (12A) will "max out" a 240V/15A circuit when you account for the 125%. Another common size is 20A, but three will overload a 240V/20A circuit, so there's no benefit to going 20A.
The rules generally prohibit putting multiple receptacles on circuits larger than 20A. If you're feeding a bunch of machines off a 60A breaker, nope.
240V on the 15A circuit
If you are in a dwelling unit, they do not allow 240V circuits unless absolutely necessary, i.e. A load is over 1440W (the max for a continuous load on a 120V/15A circuit). As discussed, these power supplies are exactly that so they don't qualify. But that's OK, you have plenty of power in that 60A split-phase service.
If you are not in a dwelling unit, you can do as you propose: pick a circuit, change every load and receptacle to 240V, mark all the white wires with tape to designate them as hots, and punch them down to a 2-pole breaker rated 15A. You could then plug up to two of these PCs in 240V mode. You must use a 240V receptacle and get 240V cords for the PCs. Do not under any circumstance wire 240V to a common 120V receptacle, that would create a booby trap for anyone else who tried to use the receptacle!
These receptacles are also made in "doubles" like the normal double receptacles you're used to (with the blades different obviously).
60 amps on 6 AWG wire
This is OK. 6 AWG wire is good for 55A at 60 degrees C temperature, however since they don't make 55A breakers, you can go up to the next size, 60A.
Making the most out of 60A/240V with neutral
You mentioned "4 circuits" so you may have already done some of this. You cannot simply hang a bunch of 120V/15A receptacles off a 60A split; you need to have a sub-panel.
Now imagine you get an 8-space sub-panel and fill it with eight 15A/120V breakers, and use those to feed eight 15A 120V circuits. Can you do that? Yes you can, because of your neutral. You have 60A on each leg of the 240V, so you have 60A/120V on one side, and 60A/120V on the other side, both at once. It will work fine.
This will let you serve eight PCs.. That is more than what you need.
Get a panel with plenty of extra room, like a 20- or 30-space, or even 42. It costs a bit more, but they often give you free breakers. And it'll be nice if your needs ever change. Your breakers equal your capacity only because you plan to load everything to 80% of max. In normal loading you could have more than 8 breakers.
Of course, the problem with this sort of activity is the power company makes most of the money. However some power companies offer plans that give you power almost for free if you play their game right... That can change the finances of mining.