Who is the master smithing trainer?

Eorlund Gray-Mane; the blacksmith who owns the Skyforge in Whiterun.

The second best smithing trainer is in Riften. To use him, however, you must bring him ten fire salts for his miniquest. After that, he will train you.

If you're looking for a fast way to get to 100 smithing, I suggest just buying a bunch of leather strips and iron ingots from all the blacksmiths around the cities and then create a bunch of iron daggers. Doing this it took me around 1 hour and 4000 gold to get to smithing level 100, starting from around level 40.