Wash your slipcovers first, to remove any sizing or finishing chemicals -- this will make them more receptive to dying.
Dyes can require either a hot-water bath (pot on stove) or cold-water bath followed by heat setting (dried in hot dryer). You'll need dye powder/liquid enough to do the yardage you have (some of the smaller packets of dye handle maybe 3 T shirts). The instructions on the box should address this.
You can get RIT dye from many supermarkets, and dye in your washing machine. RIT has been around for decades. It's a traditional standby
/more traditional colors.
Hobby stores (Hobby Lobby, Michaels, etc) and Art stores have some different brands of dye as well. You'll find different colors, and some day-glo-type colors in these different brands.
Just follow the directions exactly, and make sure *all* of the pieces of the slipcover are in the same dyebath at the same time! Make sure your vessel (sink, cooking pot, washer) is large enough for the amount of fabric completely and well-covered by water. You may have to dye pieces more than once if you're trying for a dark color. (It *is* possible to dye in the bathtub, but you'll have to commit to some time scrubbing the tub afterward w/bleach to remove the leftover dye-stain. Ask me how I know)
One of my guy's coworkers TIE-DIED his slipcovers, in shades of purple and magenta! They came out really wonderful. I don't know the brand of dye used, but they were done in large containers outside, then the dye was "set" in the drier (heat).
Along these lines, if you want to make "designs", you can mix up the dye liquid and use squeeze/spray bottles to apply the dye in patterns, swirls, etc. Recommend doing this outside on the grass,
You might want to pick up a couple of pillow covers or other small pieces of that same fabric to test with before committing the entire slipcover. If you're willing to experiment, you can dye with 2 or 3 different colors for a shaded effect, do tie-dye, half-n-half, all sorts of things.