William Rufus was the son (born 1056) of William the Bastard and gained the name Rufus after his death on account of his red hair. On his father's death, he became King of England, whilst his elder brother, Robert, inherited the dukedom of Normandy.
Crowned on 26 September 1087, he was immediately challenged by a rebellion in favour of his elder brother. William made broad promises to the native English, won them over to his point of view, and then "forgot" some of the things he had promised.
In his early years he fought in Normandy and against the Scots.
He had ill-luck with his Archbishops of Canterbury -- the excellent Lanfranc died of fever after 2 years of William's reign. The post was then held vacant, with the revenues going into William's coffers, at the suggestion of the royal chaplain, Flambard. Which meant that when William fell ill in 1093 he lacked an Archbishop to shrive him. Hastily Anselm, Abbot of Bec, was appointed, reckoned a noble and saintly man. Unfortunately he incurred the king's displeasure almost immediately when his contribution towards another Normandy campaign in 1094 was deemed miserly. They clashed and Anselm eventually retired to Rome in 1097 (which meant William took the revenues again).
He was more lucky over Normandy. In 1096 Robert decided to take part in the First Crusade. For which he needed money. Of which William had a quantum. So Robert put the duchy up as security for a ready-cash loan, and while Robert was away, William went to war there again, to preserve the family inheritance.
The King, and an arrow, intersected. Of those hunting, it was Tirrell who made a swift exit to foreign parts. Which led to him getting blamed.
On the other hand, since William had never married, he had no lineal heir (unless you count the unsubstantiated rumour that one Berstrand, an individual seemingly otherwise unknown to history, was his illegitimate son) to get stroppy over the demise, and between the Norman nobles, who disliked William Rufus because he was an efficient king, and the English who disliked him for that reason and because he was a Norman, no-one really objected when the third brother, Henry, rode post-haste to Winchester, seized the treasury, summoned the Witan and had himself selected as king (Henry I). To seal his popularity Henry also imprisoned Flambard (whom many blamed for some of Rufus' more savage efficiencies), recalled Archbishop Anselm, and married Matilda, daughter of Malcolm III of Scotland, and directly descended from the pre-Conquest king, Egbert.