This is a quest given by Stannis Baratheon with the following prerequisites:
- The Reach has declared it is backing Renly for the throne.
- You are a vassal of Stannis.
Once those requirements are met, speak to Stannis and a unique dialogue option should be available to start the quest. This requires you to have Renly Baratheon as your prisoner, which can be achieved in one of two ways: through battle (either by defeating him directly or defeating a foe/garrison that has him held prisoner), or by sneaking into the prison of wherever he is being held at the time to get him out. Either way, you're claiming your right to execute him. After having done so, make your way back to Stannis to finish the quest,
Reward: 5000 experience
Other effects: A few Stormlands lords defect to Dragonstone
N.B. in version 6 and above, you cannot directly capture him through battle. However, if you defeat a lord who has him as prisoner or break him out of prison, you are able to capture him.