Please don't do any of this without having a word with your landlord first. At the very least it's just common courtesy, and at worst you may be in breach of your tenancy agreement, even if you think he'd "never know."
Which is why I suggested he do exactly that.
Chasing cables is down to the build, sometimes it's rediculously easy, sometimes it's got some tricky bits. In terms of terminating you have several options.
Firstly, don't covert any existing sockets, especially electrical ones. Instead, fit new boxes. I use standard metal back boxes and combine this with shallow surface mount boxes for extra cable play, though I've chased three cat6 cables into a deep back box and surface mount plate with two connected and one for redundancy. In fact that's for my main PC and it worked lovely.
I buy a lot of my gear for this sort of thing from a shop that rhymes with BrewTix, its a chain that's everywhere, not always the cheapest, but usually has been for me, shop around.
You could use those brushed cable entry boxes, I think they're ugly and awful so I'd terminate into proper RJ45 plates with a punchdown. I bought a proper Krone tool but there's cheap alternatives. It's actually very easy.
When I wired our lounge and conservatory I had never run ethernet before, it only took a little YouTubeFoo before I was confident.
Going up through floors is always trickier, but often you can get up behind a wall and come from there, it will take some investigation. If you're not super confident get a sparky to take a look and run the cables, you can do all the terminations yourself to save money (sparks I've met don't tend to do it anyway).
Basic tools you'll need are an electric Drill with hammer function, masonary bits, punchdown tool and suitable wire strippers. Parts you'll need are Cat5e/Cat6 cable, Backboxes (either surface mount or proper ones), faceplates (RJ45 or brushes).
If you're not confident that's cool, we all have things that we aren't comfortable with. Mines plumbing, I hate it with a passion, I've never sworn so much as when I fitted the dishwasher.
You don't have any NTL boxes around do you? I discovered here they often used cat5 for telephone wiring and I've managed to piggy back onto those for a connection upstairs.
Long post, sorry.