There are a lot of neat effects when you use high voltages with water. See this video for an example. In this case, the voltage is applied across two different beakers. Watch carefully and you can see that the water does rise up the slope of the beaker with a high enough voltage. One important detail is that this water has to be very pure; ionic impurities such as in salt water would cancel the effect. This is because the ions act as mobile charges, making the water conductive. What would happen is that the mobile charges would crowd around the oppositely charged electrodes, creating an opposite voltage, and cancelling out any effect. This is the same theory as to why you get current when you put a voltage across a metal conductor.
Why does this effect occur? Across large voltages, it's believed that a larger portion of the water ionizes itself into OH- and H+. These are the charges that are providing the electrostatic attraction when the water connects. Why exactly it forms a water bridge is still somewhat mysterious. Evidently, the charges aren't completely mobile. Otherwise, after the water bridge connects once, the charges would transfer and the water bridge would break. This video provides a possible explanation to what happens.
Your question also touches on an idea called Debye screening. When you have ions dissolved in water, the surrounding polar molecules tends to effectively reduce the electric field around the ions. This image somewhat illustrates this idea.
If your fluid had a net charge, rather than climbing up the walls of the container, the charged molecules would just tend to orient themselves near the outer surfaces of the fluid. The repulsive force wouldn't be strong enough to defy gravity. There are however, surface tension effects that make water drops on a surface form into beads rather than sitting flat on a surface, or climb up glass surfaces. This is an entirely different effect however, and is more an interaction between the polar water and a surface.