I bought netgear pl 1200 powerline adapters for my Internet my modem and router are in my basement my gaming console is in my room on the 2nd floor, 3rd floor if you count the basement. They say for best results to use outlets on the same circuit. I just need to know if I can wire a new outlet in my basement and connect it to the circuit breaker my bedroom is on, so they can be on the same circuit for best results. And do so without running wires through the walls to my room.

Hard to give you an answer since we have no idea as to how your house is wired, what type of breaker box you have or what type of breakers. A breaker that accepts two wires will have ridges or grooves or a retaining washer that holds two wire ends in place fully under the screw head without wrapping them around the screw shank and just before the screw is tightened. That is a permitted way. Better to check the name of your box and the name and listing of the breaker and then compare it to a similar breaker. Many Square D breakers have 2 wire connections. You also need to check with your local codes to see what they allow. Some allow pigtails to a breaker, others don't.

The chances of your bedroom being on the same circuit as your basement are almost zero. Find the breaker that kills receptacles in the basement, not necessarily the lights, and switch it off. Now see if your br receptacle has power, likely it will.

They say for best results to use outlets on the same circuit.
I suspect the objective of this is for all of the equipment to be on the same half of the phase.

Breakers 1,2,5,6,9,10,13,14,17,18, etc. are on one half of the phase.
Breakers 3,4,7,8,11,12,15,16,19,20, etc are on the other half of the phase.

If basement and bedroom are on the same half of the phase you don't have to do any thing.
If basement and bedroom are not on the same half of the phase, move one of them up or down one space and they will be.

It some situations you have two circuits on opposite legs sharing a neutral. Be advised that if your panel is wired in this configuration you just can't move breakers around. You may be overloading the neutral(s) if you do. Just be aware of what you do.

Sorry, I but disagree. If you have a MWBC (two circuits sharing a neutral) and you move the breaker up or down one space you haven't changed any loads. You have only reversed the half of the cycle the loads are on.

Sorry, I but disagree. If you have a MWBC (two circuits sharing a neutral) and you move the breaker up or down one space you haven't changed any loads. You have only reversed the half of the cycle the loads are on.
If it's a double breaker, that's correct. But there are many single pole breakers used in MWBC configurations... and I know that shouldn't be done.

Sorry but I disagree again. Nothing changes but the half of the cycle it is on.

PS
I have about 20 MWBC in my home.