How can the refractive index of water in a glass bottle be determined?

Perhaps the easiest way to determine this would be by using Snell's law.

This requires that you shine a beam of light incident at some angle to the normal of your bottle (let this angle be "theta 1", or Th1). By then observing where the beam of light is refracted to once inside the bottle, you can then measure a second angle to the normal, this time inside the bottle (Th2). You can then use Snell's law to equate n1*sin(Th1)=n2*sin(Th2), where n1 and n2 are the indices of refraction of your two respective media. Since the index of refraction of air is known (n1=1.0002926 at standard conditions, it is often approximated as 1), and the two angles are known, you can then solve for the remaining unknown, n2, the index of refraction of water.

It is likely that for best accuracy you will want to test this at various angles of the incident light, which would then allow you to take an average of your calculated values for n2, which is likely more correct than a single measurement.

Light will be refracted by the glass of the bottle, but luckily the light leaves the glass at the same angle that it entered it at, and the refraction effects at both sides of the glass interface cancel out and can be ignored in calculating the index of refraction of water