When you first start Mount and Blade II: Bannerlord, Fame is something you need to start a Clan, but from there you will need to continually acquire more to expand your prowess and make a name for yourself. Fame is important to maintain, but it's a slow process to give your character more. What are the best ways to get fame?
Fight against looters and bandits
The easiest method is to attack and continually fight any gang of looters or bandits you see roaming the area. Now, you don't always want to go for the small numbers. For example, if you are running with an army of 30-40 soldiers and attacking a small gang of raiders consisting of five to 10 attackers, you will not gain too much fame. You earn a little, but not enough to count.
You want to use smaller troops in your army to increase your fame. If you roam the lands with an army of 15-20 soldiers by your side, you receive more fame for attacking similarly sized raiders, or for attacking larger groups than your own. You'll need to make sure the troops under your command are strong and that you have a set of essential skills to help them and turn the tide, but it's likely that a slightly larger group won't stand a chance. Do not walk around the game with the maximum number of troops under your command. Fight with a smaller force and reap the rewards.
Take part in tournaments
Some of the towns and castles you visit will have a tournament that you can participate in where you engage in combat. For those who wish to increase your fame in this way, you need to make sure that your character has the proper skills and equipment to fight. You mostly want to focus on being good with a one-handed or two-handed weapon. A bow or crossbow will work fine for a little while, and it won't help if your opponent rushes you and is still alive. If your focus is on guns, you probably won't last too long as it's not the best in close quarters combat. A sword and shield are probably your best bet, especially if you have mastered the technique of shielding.
Become a mercenary for another faction
For those who have already created the first level of your clan, you have the option of offering yourself as a mercenary to another faction. It's a great way to gain new relationships with a faction, earn money, and you can earn easy fame due to the importance of battles, especially if the faction is at war.
To become a mercenary, you want to find a notable figure in this faction. You can do this by walking around the map and holding the Left-Alt button, showing the names of the stray characters you see scurrying around. If you see an icon with a specific name, rush over to them and ask, "There is something I would like to discuss." From there you now want to ask, "I would like to enter into the service of" and the end of the sentence should be the one ruling over that particular NPC. You want to offer them your sword for the right amount.
When you do this, that faction should turn green for you throughout the game, and any factions they are fighting should turn red. You can now freely attack raiders, enemy factions, and aid other allied factions in combat. Becoming a mercenary gives you more opportunities to gain fame apart from only encountering looters or bandits in combat.
You want to engage in combat as often as possible if you want to increase your fame. Keep your blade sharp and your arrows true if you want everyone to know your name.