How Did House Corrino Come to Power? Corrin, 88 BG

House Corrino came to power through the Battle of Corrin in 88 BG, which gave the family both its name and the rule of the new Imperium. It then held the Golden Lion Throne for roughly ten thousand years, until Paul Atreides forced Shaddam IV to give it up. (Dune; Dune Wiki)

How did they win the throne?#

The Dune Wiki says the Battle of Corrin in 88 BG granted the Corrinos their name and supreme rulership of the new Imperium, and that the Golden Lion Throne honours the House's crest. Frank Herbert's own glossary is narrower: it calls Corrin a space battle near Sigma Draconis that confirmed the supremacy of the House ruling from Salusa Secundus. (Dune; Dune Wiki: House Corrino, Battle of Corrin)

What did the long rule rest on?#

The Sardaukar, a large vote in the Landsraad Council, and a willingness to be quiet about treachery. The wiki says the House held a vote larger than any other Great House, and repeats the claim that only all the Great Houses together could beat its Sardaukar. (Dune Wiki: House Corrino)

How did it end?#

Complacency and corruption weakened the Sardaukar, and Shaddam Corrino IV made poor decisions while the Bene Gesserit schemed, so the Atreides gained standing. Shaddam ordered the Atreides onto Arrakis so House Harkonnen could destroy them. Leto was killed, but the Fremen uprising under Muad'Dib caught both the Corrinos and the Harkonnens off guard. After the Arrakis Revolt and the Battle of Arrakeen, the Corrinos gave up the throne, and Paul's marriage to Shaddam's eldest daughter Irulan legitimised Atreides rule. Shaddam and his family were exiled to Salusa Secundus. (Dune Wiki: House Corrino)

Short timeline#

1. 88 BG: Battle of Corrin. The ruling House takes the name Corrino. (Dune; Dune Wiki)
2. About ten thousand years of imperial rule. (Dune Wiki)
3. Shaddam IV sends the Atreides to Arrakis. (Dune Wiki)
4. The Fremen rise, the Battle of Arrakeen follows, and the throne passes to the Atreides. (Dune Wiki)
5. Exile on Salusa Secundus. (Dune Wiki)

For how Paul got there, see how Paul became Emperor.

My read: Herbert gives you the outcome and the name, and the later books fill in a war story around it. Use the glossary for what Dune says and the expanded novel for the colour, and don't blend them.