And how was this connected to the rise of the Ummayeds?

I've tried reading it on Wikipedia (which may not be reliable on such a religiously charged topic), but honestly, it's a bit confusing.

So, from what Wikipedia tells me, I think what happened is something like: Some people murdered Uthman. I'm not sure why. Ali was chosen as the fourth Caliph. He didn't punish Uthman's murderers (again, I don't know why). Muawiyah, Governor of Syria (and a relative of Uthman?), refused to accept Ali as Caliph because he hadn't punished Uthman's murderers (but maybe it was really because Ali's capital was in Iraq, not Syria?). They had a big battle, and after three days decided to solve it peacefully. Ali chose people who were involved in Uthman's death (why?), but others didn't accept his choice. The arbitrators decided that Muslims should elect a Caliph. Ali didn't accept this. Some extremists on Ali's side were fed up with negotiations, saying Ali wasn't pious enough to fight properly, and went off and made their own faction. Wikipedia seems to say these guys are the same ones that murdered Uthman? The war continued, but the extremists decided to assassinate both Ali and Muawiyah. They killed Ali, but I assume failed to kill Muawiyah? And then Muawiyah made peace with Ali's son Hassan and became Caliph.

How accurate is that? What's the deal with Ali failing to punish Uthman's killers, did he just not know who they were?