Dyeing your hair is just as easy and bleaching it. In fact, it's probably easier because you don't have to worry about timing (you'll see what I mean), roots, etc. These directions are only for dyeing your hair using semi- or demi-permanent dye, though, so if you're using permanent, follow the instructions on the box.

How to Dye Your Hair

Preparation

Basically, you have to do the same stuff you did for bleaching. Cover your bathroom, get gloves, old cruddy clothes, the whole nine yards.
Slather vaseline or baby oil on your ears, your forehead, your hairline, your neck, and any other place that the dye may touch.

The Process

Make sure you have your gloves on, or you are going to have dyed hands for awhile.
Squirt the dye into a non-metallic container if it's in a tube. If it's in a jar, you can just leave it in there.
Take your tint brush (or your hands), and start applying the dye. The tube will say do not get on scalp. This is pretty much impossible. You're going to get some on your scalp, but don't worry, it will wash off after one or two shampoos.
Make sure you've gotten every spot on your head. You might want someone to check the back for you to make sure you've got it all.
If you have a processing cap, put it on now. If not, don't worry about it.
Since this is semi- or demi-permanent dye, it has no chemicals in it to ruin your hair. You can leave it in for as long as you want. I recommend at least thirty minutes. I usually leave mine in for an hour to an hour and a half. I've heard of people leaving the dye in for six or more hours, but that's a little excessive. After about two hours, your hair's pretty much absorbed as much dye as it's going to.
When you think it's set, step into the shower and rinse your hair. Try not to let the water run down your face because it will stain. You may have to stay there a long time before the water runs clear. Don't shampoo if you want your color to last.
After all the color is rinsed out, deep condition your hair.
Gaze into the mirror and be entranced by your kick-ass hair color.

Clean Up

To get dye off your skin, use an astringent or alcohol. Pour the astringent on to a paper towel and scrub. You do have to scrub, believe me. They do sell actual dye remover, but I don't think it's necessary when astringent does the job.
To get dye off your sink, toilet, floor, etc. Use a cleaner like tilex, lysol, or clorox to get dye off bathroom fixtures. Again, you may have to scrub a bit. I would actually recommend doing this while your dye is setting, because it is MUCH harder to get dye off fixtures after it's been allowed to sit for awhile.
You will most likely NOT be able to get dye out of clothing. Most dyes, with the exception of the cream form of Manic Panic, stain clothing for life. This is why you wear old, crappy clothes.

Tips for Lasting Color

If you want your haircolor to last for awhile, you can add a little bit of the dye to your conditioner. This will probably stain your hands lightly, but it will help your color stay in!
Don't wash your hair for one or two days after you dye it. It's going to feel greasy and yucky, but resist the urge!
Sometimes the shampoos made for colored hair are gentler and don't strip the color from your hair as quickly.
I cannot stress this enough: CONDITION YOUR HAIR. Condition, condition, CONDITION.
Don't use hot oil on your hair because it tends to fade the color.