Take Dave and Jim’s advice. HSS only and don’t be afraid to crank up the RPM. Cleveland drills are really good. I use screw machine or stub length drills. They are shorter and stiffer. There really should be no visible runout on the drill when held in the chuck. The trick if there is one to getting a deep hole to be straight is to make sure the first part or about 3 diameters of the drill is started properly. Center punch and use a center drill. For small drills the part should be clamped down so they can not wander even a little bit. Most machines won’t spin fast enough for the recommended RPM in steel 1/16 inch HSS, 8,000 RPM. Use lubricant and peck drill that way you will clear the chips often. Chips are the major cause, after running too slow, of broken drills and drifting holes. Don’t let the drill dwell in the bottom of the hole, it will dull it and if you are drilling tool steel it will harden the steel st that spot and there is a possibility that you will not be able to continue drilling unless you have a carbide drill. So always take a chip and slow down your feed when you’re starting to break through the metal. Another thought. Drills tend to cut oversized hole. One reason is the point is ground uneven making the small chisel point off to one side, another reason to buy quality drill bits.
Hope this helps.
CheersRichard