his writing was criticised by another playwright in 1592, so presumably his name was already known at that stage as both writer and actor:
From Wikipedia:
It is not known exactly when Shakespeare began writing, but contemporary allusions and records of performances show that several of his plays were on the London stage by 1592. He was well enough known in London by then to be attacked in print by the playwright Robert Greene in his Groats-Worth of Wit:
...there is an upstart Crow, beautified with our feathers, that with his Tiger's heart wrapped in a Player's hide, supposes he is as well able to bombast out a blank verse as the best of you: and being an absolute Johannes factotum, is in his own conceit the only Shake-scene in a country.