You do not need a gaming keyboard or mouse to successfully play this game. I personally prefer a mouse with many buttons, but a two button mouse is certainly adequate.

I would recommend you learn how to move with the mouse. I have 'A' and 'D' rebound to strafe left and strafe right, so the ONLY way my character can turn is via holding down the right mouse button and moving the the mouse. That solves the problem of having to change your hand position to strafe at the cost of having to mouse turn. I highly recommend this, as keyboard turning in MMOs is too slow for my taste; that said, people play this game on PS3 controllers all the time, so that's not a necessary change either.

As for what buttons I have for skills. I use 1-5, q, e, r, t, f, g, h, z, and x, plus shift versions of the above in order to use my skills, which is 28 key combinations, which is enough that I don't need to click anything or change hotbars. Since I DO have a mouse with extra buttons, I substitute some of the more awkward combinations with some things bound to mouse, but you could compensate by using the control key.

I do HIGHLY recommend you get used to NOT clicking things, though. Sure, you probably don't use Cover very often, but when you do I'm sure you'll want to use it NOWNOWNOW, and clicking an ability is almost always slower than a keybind.