That’s a huge task, with many 1000s of scientists going back some 200 years, so you’ll need to track all that down yourself.
Most climate scientists either work for gov agencies, like NASA or NOAA, or teach & do research at universities. They just basically get their salaries. If they get grants for their climate research, that usually pays only the portions of their salaries (not above their salaries) to give them course releases, plus funding for equipment, research-related travel, and research assistants.
Here are some helpful links to get you started on the original climate science & scientists:
The Discovery of Global Warming at
aip.org/history/climate/summary.htm
The History of Climate Change Science at
en.wikipedia.org/wiki/History_of_climate_change_science
Look especially at the info on Fourier, Tyndall, and Arrhenius.
As for current climate scientists you could read the IPCC reports and see the 1000s of scientists they site then check up to see where their money comes from.
As mentioned it is many 1000s.