This is not Power Over Ethernet (PoE) which is a way to power devices with Ethernet cabling and requires specialized power supply and equipment that support being powered this way. What you're describing here is Power-line Communication (PLC) colloquially "Ethernet over power" and given the marketing name of HomePlug Powerline.
Depending on how your wiring is, it can either work very well -- or not at all. It depends on distance, connection points, and so on. The more direct the path, the better. For instance, adjacent rooms on the same circuit can work great. Going across multiple junctions or physical points in a circuit breaker can dramatically reduce throughput. Naturally older wiring or noisy electronics on a circuit can also greatly reduce throughput. Also Powerline adapters typically don't work when plugged into a surge protector -- a whole house surge protector may also defeat them.
There's also huge security risks -- especially in multi-family homes or where you can easily access an outdoor power receptacle. If you use a default configuration, it's trivial to get onto the network as a rogue device.
Another, possibly better option if your home has rooms wired for cable is to use MoCA adapters instead. These basically allow you to run Ethernet over the coax ports used for cable. These are much more reliable and less susceptible to interference than Powerline. You can also get MoCA Wi-Fi repeaters.
Running Ethernet cabling is really your best option. It's not really hard to do if you're even remotely handy but make sure you use cabling rated for in-wall use and check your local electric code (don't be afraid to call your city/county inspector's office with questions about this) for gotchas like "must not be within X feet of high voltage wiring" and such. I ran all the rooms in my condo for Cat6 through the attic.
The supplies you need for this project are a long drill bit (6-12") and drill, fish tape, a small drywall saw, "old work" wall boxes, CM rated Cat6 cable, RJ45 ends, wall plates, RJ45 terminal blocks, and a RJ45 crimping tool. The tech stuff you can get from Monoprice for a reasonable price (don't get Monoprice wall plates though, they really suck) and everything else you can get from any hardware store. If you're lucky and your rooms are already wired for cable, you may be able to run the Ethernet cabling through the existing holes drilled for cable and then you just need to change out the wall plates. That'll save some drilling headaches.