There is nothing really special about critical temperature at any given pressure.* So I don't think there is a direct experiment to look for critical temperature. You can, however, look for critical point which will give you both critical temperature and critical pressure. You do this by following along the transition point until the latent heat of transition goes to zero.
For example, you can use a constant pressure calorimeter, supply heat at steady rate and track temperature change. Look for the place where temperature temporarily levels out - that's your transition temperature. Increase pressure and repeat. The plateau will now have narrower range in energy. Keep increasing pressure until that plateau disappears. That's your critical point.
* Edit: Actually, there is an estimate you can make based on surface tension and Eotvos rule
. Surface tension at critical temperature does go to zero, so I guess, I shouldn't say that there is nothing special. I suspect, it's very difficult to measure the surface tension near the critical point, but you should be able to at least extrapolate the approximate critical temperature by taking measurements at different temperatures. If nothing else, this will tell you where to look for critical point in finer detail.