What's the best way to get a coaxial cable to an external HDTV antenna?

"Do I just drill a hole, feed it through, and then caulk it to seal it up?"

Yes, from the basement out. I usually use a 3/8 inch drill, about 16 inches long.

"Is it better to drill down, get the cable into the basement and then go outside from there?"

Yes, Drill down then drill out just above the foundation. BUT... before you drill either hole, check and double check so that you are not going to drill through a pipe, electrical wire, etc. on the other side. Also, make sure you don't drill through a carpet (you could damage it), pull up the edge of the carpet and THEN drill.

That really depends on how you define "best." The easiest way is drilling a hole in your floor and running the wire through the crawl space/basement to the outside (assuming you are on the first floor and have a crawl space or basement). If you want the best look (ie wall plate, limited amount of wire exposed on the outside of your house), it's going to be a little more involved, especially if you aren't on the first floor. I actually do cable/telephone installation for a living, so if you could give a little more detail about your house and the look you are going for, I'd be more than happy to offer you more detailed advice.

there isn't any converter for this challenge. in the experience that your television would not have a RF co-ax connector, it would not have a television tuner in any respect, meaning it can't technique the sign from the cable-from-the-wall. you will could desire to call your cable service and get a HD cable container, then connect that to the television with HDMI.

drill Straight out.

This sounds like a question for a contractor. You might want to move this question to Home & Garden.