How would you clean out a washing machine after using it to dye clothes?

Run it through the whole cycle with bleach.

What @chyna said. Maybe sponge some bleach down the sides, on a sponge, first. Maybe with some old towels that you don’t care about.

Dye doesn’t usually adhere to metal much so I’d think maybe just running an empty load with a little white vinegar or bleach would be fine. Or run a wash of dark clothes like jeans and other black items. Just don;t combine your dyed clothing with anything not the same color in the future. Also, run your freshly dyed clothes in a wash with white vinegar as well to help set the dye.

I used to hand dye items in my stainless steel kitchen sink and never had an issue woth staining, same with enamel just bleach the sinks out when done.

My wife uses Color Catchers for washing her just completed quilts. Two or three per load, use them with the first dark load to capture any dye still on the pipes.

I would do a bleach cycle and then another cycle with a crap towel to test it before putting in a regular load of laundry.

The bleach is good, but I find that after washing new jeans, lysol or clorox wipes help get rid of the dye pretty well without me having to waste a whole tub of water.

I use this stuff after my dye projects. It does a good job of cleaning up the dye and doesn’t waste the water that running a full load would. Just do a normal whites load afterward to make sure the bleach rinses off before you wash your brights.

If I was going to use bleach, how much should I use? And would I need to worry about some of the bleach hanging around afterwards?

I would use one cup of bleach. I wouldn’t worry about some of the bleach hanging around, but you could wash some towels as your next wash just to be sure.

If you use the bleach method make the next load you wash one of whites to be extra sure.

I hate wasting stuff; be it water, energy, food, etc.

After the dye project I’d make sure I followed up by doing a dark load like jeans, black socks, dark towels, etc. I figure that would clean it enough.

I’m with @chyna all the way.
It is what I’ve done, exactly.

When you run the load with the cup of bleach, you can always wash things you don’t care about, too. If you have old clothes with paint or bleach stains on them, old towels you use for projects, or a synthetic blanket.

Follow the directions on the package of dye. Or do a full load (maximum water level) with bleach, as others here have stated. Then wipe out lid and any other areas not submerged by water with a rag soaked in a solution of bleach.

I would do what @LuckyGuy said. Throw in a load of black or dark blue jeans, dark towels, dark sheets and call it a day.