It's a pretty straightforward procedure. Took less than an hour. I replaced the 500GB Seagate Barracuda with a Seagate Barracuda 2TB, model ST2000DM001. So far, after about 30 minutes of use, it's working great, and I haven't had the fan problem, but we'll see what happens.
Kevin's advice about the tweezers for extracting and inserting the screws that attach the display to the case was right on (although I used a small pliers instead).
Before you begin the replacement procedure, I highly recommend putting the new drive into a hard drive dock, and restoring to it from a Time Machine backup. Try booting off the new drive once to make sure it works. Then, when you swap the new drive for the existing internal drive, it'll be ready to go with no further effort needed.