It’s only cheaper if you don’t royally screw it up and have to re-do it a handful of times before ultimately going to a pro to fix your mistakes. I’ve been coloring and cutting my own hair for almost 18 years, everything from lime green mohawks to long layers with blonde highlights… and I don’t even know that I would want to risk a complicated color like this, let alone if I didn’t have experience coloring my own hair at home.
Otherwise, you can try doing an all over color with the purple/black (assuming it is dark enough to cover what you already have, and leave it however many inches that you’re hoping to make blonde from your ends. Trying to lift color is difficult, especially a dark color. You’ll almost certainly need a friend to help you with this.
Then go back and try bleaching the ends. I would definitely suggest using a brush and bowl method, but work quickly so you don’t have bleach sitting on your ends forever. After a while it is no longer lightening, just destroying. You may have to bleach more than once to achieve the blonde that you want, but don’t do it all in one day, or you are likely to fry your hair and end up with a broken mess. You probably want to buy a toner to go over it, just bleaching darker hair tends to leave it a sort of orangey-yellow mess. At home blonde can be really challenging.