Shakespeare is a classic because he is the first person who really addressed such literary themes on stage; and as well was a pretty good poet. Shakespeare is only difficult to understand because langauge has evolved since them. If you understand 16th century English, he's actually pretty clear.
As well, he stood out as someone who didn't work with purely contemporary things, nor did he go into history that was quite so idealized as things such as the Grail Romances, with were practically mythology. Shit, back then the idea of wrting about, say, the assassination of Julius Caesar was radical as hell. And he didn't do it as a dry history, he added fairly fitting drama to it all.
Mind you, there are also a lot of red meat for the audience plays, so it's not like he was all classics all the time.
Then, you have to consider that a lot of the themes he wrote about still resonate with people today (at least once you get past the language barrier). That is not an easy thing to do, as a writer.
As well, Shakespeare gets my personal respect for The Tempest, as it was pretty much an early fantasy novel; it wasn't based on any particular mythology or folklore (though it invokes several), it was Shakespeare pretty much making up his own setting.