Let's face it, there's a reason people gave up those clothes to the thrift store. Sometimes it's because they're ugly. But if the ugly just has to do with color, sometimes that doesn't have to stop us.
Let's be clear: dyeing can be unpredictable. Some fabrics just won't accept the dye, or accept it oddly. If different kinds of materials were used in a single garment, part of it may take the dye, and part may resist. I would never dye any garment I actually care about.
Dyes come out truest on white fabric. If the garment is already colored, the results are more unpredictable. Frequently the final result will be darker than the dye color. It may be more muted, and it may mix with the original color.
The solution is to bleach the garment first. Rit Dye produces a color remover, but you can also use basic laundry bleach. You may not get all of the color out.
Rit Dye comes with instructions for dyeing. It suggests dyeing in the washing machine, but those directions were written for a top-loading washer. I am not at all sure if it would work for a side-loader, which is what I own.
So I use the stove-top method, and since I find that higher temperatures work better, I would strongly suggest stove-top over using a bucket. Get a pot big enough to hold the garment and enough water to cover the garment. Heat the water before putting in the garment, and then keep the stove cooking. I keep things in for about 30 minutes.
Rinse the garment thoroughly in a sink, then throw it in the washer to get rid of any trace amounts of dye. Scrub the pot clean.