How should I replace a 6 wire, 12 gauge electrical outlet?
The best way is to use a pigtail. Use a large wire nut to join four 12 awg hot wires in the back of the electrical box. One of these four wires is the pig tail that will come out and join to the outlet at the screws. Do a similar arrangement for the four neutrals (one a pigtail) and bring out the single neutral to the outlet. 

This arrangement also makes it far easier to mount the outlet after the wires are attached.

Don't forget the GND wire attached to the green terminal on the outlet.