Remove the faucet and drain, and use a putty knife or scraper to remove any old caulk from the counter or where the backsplash meets the countertop. Alternatively, you could tape around the faucet and drain instead.

Scrub the sink and counter mild abrasive cleanser (like Comet) and rinse well.  Then scrub with a hard water cleaner like Lime-Away and rinse well. Be sure to remove all soap and residue.

Sand with 320-400 gfit sand paper to rough up the surface and the give the paint something to grab. Clean the sanding dust from the sink really well using wet and dry cloths, and a tack cloth.

Tape any surfaces that you do not want to get the tile paint onto (walls, vanity, etc) with a painter's tape. You may want to use some drop cloths to protect the cabinets and floor.

Wear a good respirator (and safety glasses and gloves) while mixing the parts and painting. Follow the instructions on the can to mix the tub and tile paint. It's pretty simple: just mix Part A (applicator) and Part B (base).

Use a fine-bristled brush to cut in around the edges of the counter with the paint. (I actually did the entire first coat with a brush instead of a roller.  I found it much easier to get a nice thin coat on with a brush.)

Use a 4" foam roller to roll the paint in THIN coats on the larger areas.

Let the first coat dry. Rustoleum recommends waiting 1-2 hours between coats, and applying 2 coats. (I coated the vanity top really thinly, so I actually applied 3 coats for the coverage I wanted.)

Apply the second coat (and I did a third coat as well). The trick with this stuff is not to get a super thick layer on there, but not too thin either.  Too thick = bubbles and drips.  Too thin = weird texture and marks. Let each coat dry for at least an hour before starting another coat.

Let the paint cure. Rustoleum recommends 24 hours of curing before touching the surface and 72 hours of curing before getting the surface wet. Don't try to cut this short! Remove the tape and allow the epoxy paint time to cure.

Reinstall the faucet and drain after the recommended 3 days of cure time.

Use bath caulk to touch up the edges where the side backsplash piece meets the countertop.