Place a strip of tape (vertically) on opposite sides of your glass. Measure the distance, w, between the two strips. Set the glass on a level surface and, with the glass empty, shine the laser through the glass at an angle with the horizontal. Have the laser fixed to something so that it stays at the same angle. Position the laser so that the beam is right beside both strips of tape.
Mark on the strips the entry and exit points. Measure the difference in height: h2 - h1, of the two points. Calculate the angle of incidence θ1, by using the formula tan θ1 = h1 / w.
With the laser in the same position, fill the glass with water and mark the entry and exit points. Again, measure the height difference, h2. Calculate the refraction angle (tan θ2 = h2 / w).
Using Snell's law, n1 sin θ1 = n2 sin θ2, calculate the index of refraction for water, n2. Consider the index of refraction for air, n1, to be 1. Do this with three different incident angles of the beam. Compare your result with the accepted value for the index of refraction for water.