You need an appropriate bit for glass. Depending on the size of the hole, that would commonly be a tungsten carbide drill bit, diamond burr, or a diamond hole saw (I'm referring to this type, which you can buy in just the size you need or sets of sizes; they aren't expensive):
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To prevent shattering, you need to dampen the vibrations. When you score a piece of glass and then tap the score with the back end of the scoring tool,you don't hit it hard enough that the force of the blow breaks the glass; a light tap creates vibrations, and the glass breaks itself. Similar action causes glass to shatter when you try to drill it.
I've done a cool demonstration for kids of cutting a glass microscope slide cleanly with scissors. The secret is doing it under water, like submerged in a basin. You can do the same thing with a bottle.
Get a basin or plastic tub big enough to hold the entire bottle submerged under water. Get the bottle under water and completely filled with water. Use a long flexible extension for a drill or Dremel tool like this:
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Do the drilling under water, maintaining control of the bottle and tool, and using normal hand pressure to drill. You won't have the alignment afforded by a drill press, but if you start lightly to establish a mark, you will be able to keep the bit in the same location without it wandering.
Depending on the thickness of the glass, it may take a little time to drill all the way through. Just take your time and be patient. The water will lubricate and cool, as well as dampen vibrations. When you're done, blow any water out of the chuck with compressed air and dry the extension and bit well.