A practical way to judge the depth before dispatch is to see if the water splashes when a surface is stamped on; this test is preferably to undertaken by the First Officer or Engineer thus enabling the Captain to observe the result.
Easier still is to observe vehicles and see if they splash when moving. Another method is to see if the surface is a good reflector, if so then as above, the water depth is significant and probably enough to be considered ‘contaminated’ (3mm).
During landing it may be possible to judge a reflecting surface, particularly at night, but this is very much a last chance check or decision to go-around and reconsider.
If you can hear the rain on the windscreen during the approach then it is heavy; you will have to build experience to relate your aircraft type to the rainfall intensity.
All other advice as in previous posts.
Don’t assume that a grooved runway will be well drained; rubber, dust dirt, de-icing fluid, etc can block the grooves as can a strong crosswind dam the water on the runway with or without crowning.
Also ‘concrete block’ runways can dish in the centre or the edge sealant can dam water even in light rain.
Hydroplaning.
Running out of runway.
Safety aspects of aircraft performance on wet and contaminated runways.
Improving Braking Action Reports.
Approach and Landing.
Edit:- Links updated