Currently have five TV's and a computer. Reception from Comcast is just fair. Checked cable connection and found that there are six splitters in the crawl. Should I run all new from the outside splitter? Run one line to each of the TV's? Or what is the best way to do this project? Any help will be appreciated.

I had to use some amplified splitters in order to get a satisfactory, quality picture. If the outside splitter is not amplified I don't think it will help. Have you checked the incoming cable hooked directly to one TV to check reception? If you see the same problem using just the cable co. signal, it could be between you and the pole or underground pedestal.

Everything should be a home-run to a centrel location using RG-6 to the wall plates.

Comcast cable modems should be connected as close to the point of entry of the cable (not mandatory). But it must be before amplification takes place.

You need an amplified splitter and you must terminate all unused outputs.

If you have 6 2 way splitters and is split and resplit and resplit, the last will be worse.
You would be better off with a 6 way splitter, Are you using RG 6 ? Can you describe the splitter outside and the splitters inside(2 way, 5-900 Mhz? etc.)
Amplification should be done at first point. Once signal is weak, an amplifier will send weak signal to more TV's.