It's not the most straightforward show, is it? Here's my take on the story so far:
Fortitude is traditionally a mining community, but it doesn't appear to have much of a financial future. The Governor, Hildi Odegard, has invested heavily in a new venture: an ice hotel carved into the heart of the glacier where Fortitude stands. For this venture to go ahead it needs official approval, which Charlie Stoddart (Eccleston) was set to provide - until he became aware that a perfectly preserved mammoth carcass/fossil had potentially been discovered, exposed when part of the glacier permafrost melted away. All of a sudden, the hotel project is in jeopardy... but the next thing anyone knows, Stoddart is dead, allowing Hildi to use his first report (compiled before the mammoth discovery) to push the hotel plans through.
Meanwhile: Mine worker Ronnie Morgan is a single father caring for his daughter, Carrie, who is the one who finds the mammoth fossil. Worried about his family's finances, Ronnie concocts a plan with fellow miner Jason to try and get money out of Stoddart in exchange for the fossil (which they've dug up and hidden away in a lock-up shed). Things don't go as plan, and when Ronnie hears that Stoddart has been found dead he thinks that Jason must be involved. Panicked, he takes Carrie and attempts to leave Fortitude, but he doesn't have enough to secure a private plane to the mainland. Father and daughter instead wind up camped out on the glacier as Ronnie tries to come up with a new plan. It's noted that a vial of anti-psychotic drugs has been found, unopened, at Ronnie's house, suggesting that he's 'off his meds' and maybe suggesting a reason for his odd, paranoid behaviour.
Meanwhile again: Sheriff Dan Anderssen is (for reasons unexplained) obsessed with waitress Elena, whose real name is Esmerelda. Elena has relocated to Fortitude after spending seven years in prison for the violent murder of her ex-partner. She's currently having an affair with Frank Sutter, a Search & Rescue worker, much to the Sheriff's dismay. Elena also (possibly) had a relationship with a man named Billy Pettigrew, who was recently killed on the outskirts of town. Pettigrew (I'm hazy about this bit) was a geologist analysing data connected to something in the glacier (oil deposits?). He was mauled by a polar bear, but cause of death was actually a gunshot to the dead. Elderly photographer Henry Tyson believes he was the person who fired the shot, even though he was trying to shoot the bear, not Billy.
Speculation: There's one important plot thread that's suggested but has no actual facts to back it up. It's possible that the mammoth carcass is riddled with some ancient disease that has become active again (causing the young boy Liam's illness) but this could just be a weird red herring to give the show a horror/supernatural hint.
Interesting stuff: The very first scene in episode one is where Henry allegedly shoots Billy. However, Sheriff Dan is already present, with a rifle, and the split-second gunshot angle leaves it unclear whether Henry or Dan actually fired the killing shot (Henry could simply have missed, and the audience are fooled into thinking he's the killer).
Also: The last time we see Eccleston alive he's just arrived home in the afternoon, ready to cook dinner that evening for a couple of researchers he's working with. He hears a strange noise upstairs (an alarm? A woman's cry?) and his dog barks. End of scene. The next time we see Eccleston, he's been attacked. In a later episode it's suggested by DCI Morton (Tucci) that Eccleston was killed that evening in the middle of preparing dinner, but this may not actually be true. What we DO know is that when Eccleston's body is discovered, Sheriff Dan is again on the scene already, this time in a locked house.
And lastly: We don't know for sure that the prescription meds belong to Ronnie Morgan, they could actually be for his daughter Carrie. And, maybe, Ronnie hasn't whisked Carrie away because he thinks he's connected to the murder via Jason; maybe he thinks/knows Carrie had something to do with it. After all, Eccleston looks to have been killed in a frenzy (with a potato peeler?) rather than in any calculated manner, so it could have been a child who did it.
There's a hell of a lot that hasn't been uncovered yet. And we have no idea who the feeder guy is (outside of being married to the doctor's daughter) or how he's potentially involved in the overall plot, so don't worry, you've not missed anything there.
Hope some of that helps - it's a great show, so stick with it!