I have done this several ways. The most reliable way I have found is to have a drill bit turned down in to a step bit first. Then weld it to a rod to make it longer.
If you are drilling a 3/8" hole start with a 3/8" drill bit and turn, grind, the first inch down to under 1/4". You can buy these already made this way. The small front end portion will guide the bit and prevent it from wondering. Then have it welded to your rod. An inch is all you need for a guide. And it is fine to drill from both ends or just one. Clamp the stock firmly and I use my drill press drilling at the lowest speed possible. Heat and vibration are your enemies. I do one direction most times but have done both. Clamping and getting the stock repositioned so all the lines in exactly same is more trouble than it is worth.
Here is what a step drill bit looks like. If you get lucky you can find one already made in a size near what you need. You can always enlarge the hole to size if needed using your first hole as a pilot hole.
http://ecx.images-amazon.com/images/I/313HJ76AP8L.jpg