I do agree that Halo 4 relied too heavily on outside material to flesh the story out. I myself wasn't very phased by this since I read up on the lore before Halo 4 came out.
Anyway I can give you brief information on how the Storm came to be, and how Chief/Cortana knew who the Didact was (be warned that I don't know this 100%):
The Storm Covenant are a fraction of the mainline Covenant who were stationed on the Elite colony world Hesduros. They have been out of contact with the main Covenant for some time, and didn't know about the Great Schism. Some of the armor the Storm have represent Hesduros organic architecture, like the Zealot Elites armor, and ship interiors. The Jackals there are also a sub-species of the Jackals we see in the previous games (like the Skirmishers and Brutes in Reach).
Jul M'dama was an Elite captured by some humans and taken to the Shield World Onyx (or Trevelyan). The humans allowed him to walk around, and he ended up finding some ancient ruins (maybe Precursor. I'm not sure). A Hurogok (Engineer) then stumbled upon him, and Jul forced him into opening a portal to Hesduros.
Once there he came into contact with the Storm Covenant, and spoke with the Keep Keeper there (again I'm not sure that is actually what happened), who told him about some ruins near the Keep. Jul then explored the ruins and found a map to "The Didact's World", Requiem: knowing full well who the Didact was. So he manipulated the Storm and became their leader: going on a holy mission to find and unleash the Didact on Requiem. He was known as the "Didact's Hand".
After finding Requiem they orbited it for 2 years since it would not open for them... until the Forward Unto Dawn drifted towards it.
Finally Chief and Cortana know who the Didact is since... well Jul M'dama was at the Cryptum when it opened. He's the Elite who neeled, and the one who shouted "DIDACT!", so Cortana, or the Chief might have heard him and picked up on it. It's reaaaaally easy to miss Jul the first time (like I did). And if they didn't hear it Cortana already has extensive knowledge of the Forerunners (as seen in the Halo: Legends: Origins) so she would most likely know who the Didact was anyway.
Again I don't know this 100%. Feel free to correct me.