Contrary to popular assumption, water does not get its blue color from the sky. Any water, including swimming pool water, will always be blue because of the nature of both light and water, and how the two interact.
It's true that calm water will reflect a blue sky, but the same water will remain blue even under an overcast sky, or indoors. The always-blue tint in water is not a reflection, but a quality of light interacting with the molecules in the water itself.
The Color of Light
If you've ever looked at a rainbow made by a prism, you know that white light is made up of many colors, spanning from reds, across oranges, yellows and greens, all the way to blue. These different colors of the rainbow are filtered and reflected by materials at different rates, which gives things their color.
Water molecules absorb the colors on the red end of the light spectrum quickly, but the blue end of the spectrum travels further through water before being absorbed. Clean water always appears blue, and deeper water appears more deeply blue as more of the other colors of the spectrum are absorbed.
In addition to the natural blue tint of water, there may be other colors in pool water. The presence of certain minerals in water, or plant life like algae (if your pool is not very clean), may reflect other colors, like green. And of course, colored tiles or a colorful paint job will have an effect on the apparent color of the water.
To see how light behaves at different depths of a swimming pool, submerge a white bucket into a pool just below the surface and note the tint of the water in it. It will be a very light shade of blue, compared to the deeper water around it. The deeper you submerge the bucket, the deeper the color will be.