Is there a 3rd-party API for aviation weather forecast, which includes additional cloud information?
Short answer:

I wish this information were available, but what you are looking for is simply not available.

Longer answer:


  There has to be something out there else how would pilots know to
  climb to a destination altitude to get above the clouds?


You are looking for data that you rightly think should exist. I would that it did, but it doesn't. Thats where the art of airmanship comes in. The answer to your question I quoted above is a complex mix of years of experience reading the weather charts, eyeballing the view out the window, and networking through the radio. 

Real world weather is too hard to predict, and is constantly in flux. The only way to know the actual cloud conditions is to get out there and see it in the moment. The ground based weather observation stations like AWOS and ASOS do a decent job, but they are very limited and often wildly inaccurate. They only provide an assessment what is visible straight up.

PIREPS are the only partial answer you will get, and these are sparse and dated. A PIREP is only a snapshot of what a pilot observed at one place at one time. 

Area forecasts will sometimes include a cloud top forecast (look for "TOP FL200" or similiar). However, a forecast—especially an area forecast which often covers half a state—is vastly different than an observation.

You've already mentioned Skew-T charts, and they do have useful data. Depending on your application, this might end up being your best bet. If you want something that doesn't have a real world aviation application (e.g. for the SIM community), you could derive your sought information from this.