Yes, tournaments are definitely the best way for increasing renown, you get something like 20 renown for winning a tournament, declining the more renown you have. If you don't know where tournaments are or can't find any then go into the taverns of towns and ask the troubadour/minstrel/other poems guys (if you already have a fortified manor minstrels and such stuff are also spawning in there, so you don't have to travel around to find them) where there is a feast going on, then travel from there to the nearest city where the feast/tournaments take place. If you onw a city then throw yourself one feast after the other. And as lazyman said, gather yourself some 20-30 knights or high level companions on warhorses and look for enemy lords with 100-150 troops, or large bandit parties, then let them catch you and fight and beat them, this will also give you some nice amount of renown. Sometimes, you also get some renown if you free a regional manor of a random town/village from bandits who have infested it.
For right to rule it is best to gather all 29 companions and send them on the rtr-missions by asking them "do you know that I aspire to become king". This will give you 87 rtr if you send them all out - so, you're almost 'there' (rtr 99) if you've send them all out. Also don't forget that you can use your companions for improving relations with factions/regions, just ask them if they maybe have any connections they can use to your advantage and then send them out on the mission.