Well, first off, using a 2-flute drill is always going to vibrate badly (or well, actually ) To prevent that you could get a 3-flute drill but at that hardness you will need it to be Carbide and not cheap. Also going from 3/8" to 5/8" would have to be done in steps as the 3-flute drills do not have a lot of lead-in. That means maybe three drills.
You could try a step-drill something like the following link, slow speed and lots of cutting fluid, but I don't like the chances of success...
You could also try a 3-flute carbide insert end-mill, but it would need a rigid machine set up otherwise it will just break up the inserts?
Have you asked at a local machine shop? They may have the tooling and equipment to do it for a lot less than 200-bucks.
Generally, I will not give you the answer to your question, but I **will** guide you into discovering how to solve this yourself.