Why did Greene disparage Shakespeare?
The passage in Groatsworth has often been taken to prove that Shakespeare rewrote other writers' plays, but it is more likely that Greene objected to the simple fact that a mere player* should think he could write real verse.
Since Shakespeare had probably achieved some financial success by 1592 (this is the last we hear of John Shakespeare's financial difficulties), and since Greene died in poverty, there may have been an understandable element of jealousy involved.
The positive information we gather from Greene (despite his intention) is that Shakespeare already had a growing reputation by 1592, enough that he was seen as a threat by a formally educated professional writer.
Selected poems by Greene are available on line.
Footnotes
-
The first snob?
Greene's attack is thus the first sign of the snobbishness about Shakespeare's origins and education which has led in its extremes to the claim that Bacon, and a host of other candidates, wrote the plays.