Sounds like a half gallon would be plenty of bleach, but I'm not sure how you would get it down there either. When I had to install a system for 29 families they designed it so the water would have 10 minutes of contact time (in storage tanks).
With water pouring out continuously like you have it's hard to believe that the well could become contaminated. How was the water test done? It is common for people to take a sample from a faucet that is dirty and it shows up as a bad test when actually the water isn't the problem.
You should remove the aerator on the faucet and heat the end of the spout with a few matches (don't do this if your faucet is plastic). Or after removing the aerator wipe the outside and inside of the spout with a strong mixture of clorox.
After purifying the spout, run the water for about 2 minutes at a good stream. Then let the water run into the test bottle being careful not to touch the inside of the cap or the lip of the bottle.
I heard of someone recently who sent a test to the lab using a jelly jar that was around the house. This is a no, no. You need the offical test bottle if you want accuracy.
What I'm getting at is that there may be nothing wrong with your well and you can only be sure by having a good test sample. If you do have to shock it, I would call a company that does well drilling and maintenance for their method.

The pump and plumbimg had been drained for the winter last December and we 'opened' the camp two weeks ago. Nothing out of the ordinary, closed the faucets and primed the pump.
This past weekend we did a water test by running the water for fifteen minutes from the kitchen faucet (aerator not removed) then carefully filled the official test bottle and mailed it out the same day.
Thanks for your help.

Steve

Bottom line....you cannot shock an artesian well, which is exactly what you have.

The bleach will be expelled from the well too quickly to be effective unless you use fairly massive amounts of it. I think that you should try to determine if the water itself is the problem or your system is. Since water free-flows out of the well casing, take a sample of that water and have it tested.

If it tests OK, then you need to cleanse your water system. In order to do this, you would have to pump bleach down to the bottom of the well while taps are open in the kitchen, bathroom and outside to cause the pump to come on while the chlorine is being injected into the well. That way, the pump will suck in chlorine and water and force that mixture into the pressure tank as well as all the piping. Once you can smell chlorine coming from all the faucets, then shut the taps off and leave it overnight. Use the water for a few days to allow the chlorine to get flushed out of the system completely and then test the water from a sanitized tap again. At this point, you should be getting unchlorinated well water.

IF...........after having this water tested again under the most scrupulous of water gathering procedures, the water tests contaminated; then you are going to have to install water purification equipment to protect you, your family and guests.

Chlorinators or UV lamp units are the most common methods. While neither is cheap to buy or maintain, your only other alternative is the constant boiling of water and that's a huge pain.

I know little about the chemicals used for pools, but perhaps chlorine disinfectants are available as tablets or other solid form. That should sink in the water. But if the aquifer feeding the well has the coliform, it will continue to have them. During childhood, I frequently had to clean out animal manure with pitchfork and shovel. Chickens, pigs, cows, goats. I am sure that we become immune to most of these bacteria from such exposure.

You mention a pump, but also a continuous flow of water out of the casing. Is the pump at the bottom of the well or the top? A couple of things and it has been a few years since I've run some small water systems....if I remember right the casing should be more than 12" above ground level...I think it's 2' but not positive. Also, I'm pretty sure the casing should be closed at the top to prevent surface contamination from getting in.
It's not real likely that this type of system can get contaminated...at least the well portion. If there are farms nearby with manure pits that can get into the ground water.
It sounds like you are going to have to find a way to clorinate all of the piping in the house. You might have to be creative. One method I saw used was to put a tee in the pipe under the home, with a shut off coming from the well. They turned off the water with the shut off, drained water from the home through the tee and then poured clorine into the tee. They closed off the tee and turned the water back on. This forced the clorinated water through the piping. The problem is that the clorinated water doesn't stay in the system very long and the contact time really should be about 12 minutes. Maybe you can come up with some other creative idea.

It is an above ground pump (shallow jet).
At this point I think I will test the well water at the well to determine the origin of the coliforms..well or system.
If the well is positive I am thinking two things:
1) I could use an older expantion tank with a sight glass and install an air fitting to it. Weight a garden hose and lower the hose into the well, then fill the tank with the bleach solution, charge the system with air and force the disinfectant into the well.
2) Would #1 be worth the effort? It is a 30ft well with at least 5gpm flow and if the coliforms are in the ground water, the bleach would be gone rapidly and the well contaminated immediately after.

I think your idea of trying to determine if the problem is in the well or the house piping is a good one. No sense going crazy until you determine where the problem is.
If it is the well, I don't think the method you describe will allow enough contact time to do what it is supposed to. If you determine it's the well, do you have a local "extension service" that gives free advice. If it's the well, there would be a good chance that others in the area might have the same problem. I would ask around there has to be some inexpensive or free service that can give advice on a well problem.