There are two parts to activating a kill right. The first stage is to make a kill right available. If a player was to attack you in Empire space without cause, you will earn a kill right for 30 days. You now have four options, do nothing, make it available to everyone, a specific group, or a specific pilot. This can all be done from the character sheet Alt+A.
Once a kill right has been made available to you, your corporation or the public you can activate it on a pilot in game. Simply click on the pilot (on your overview), and then click the cross-hair button on the top right corner of you selected items window (Just above your overview on the top right corner of your screen). You should then be able to activate the kill right if you’re willing to pay any applicable fees.
For more information there should I think be a tutorial video in the in-game flight academy. Press F12, go to the tutorial videos tab and look for kill rights under crime watch.
If you need any more information, don’t be afraid to ask or chat to us in the Rookie Help Channel.