This is something I have to do professionally at times, when I'm sampling water from a well, sometimes as small as 2-inch diameter, sometimes as deep as 150 feet, when the purpose-built bailers didn't make it to the site (like when I land in the Aleutian Islands, but my luggage doesn't).
Kevin's stick and bottle is very direct and effective. A future tweak – use a shear lashing – those wraps between the bottle and stick REALLY snug it up to the stick:

If you KNEW* you'd be getting water from far below, here's the real deal:
Those bailers have a check valve (a ball check) at the bottom, so they fill, but don't empty through the bottom. Those 3 mid-sized ones in the photo hold just over a liter each (so one can fill a one-liter sample jar with each bailer-full). I've made mini-bailer by cutting off the bottom, business-end of it and poking a new holes for the string.
Gotta get the eldest child from school. More later. . . .
Just got roped into assisting a PE teacher in running an orienteering course for 60 5th graders.
Back now. Anyway, options, continued:
With any plastic bottle, you can tie a rock, that weighs more than the full water bottle will, to the bottom of the bottle. Lower it, wait till the bubbling stops, and then retrieve it.
Yet another approach is to poke a small hole in a bottle, add a weight, and give it a long time hanging on the string to fill. Then retrieve it quickly, before much water drains from the hole.
And, for those of us who travel with tubing and check valves (UL of course): put the check valve at the bottom of the tubing, lower it beneath the water surface, then jerk the tubing up and down quickly. The inertia of the water will cause it to stay up while the tubing is pushed down and the check valve will cause the whole column of water to be lifted up when you lift the tubing. This is good to over 25 feet and can be done through well as small as 1" diameter.
When it gets really ugly, you lower your hankerchief into the water, bringing it up, and wring it out repeatedly. (Kayak bailing sponges work much better).
*I've kayak-camped on a Mexican beach that had a well that had to be bailed to get the water to the surface.