To be gunning for someone means to try to harm someone or damage their reputation. Another meaning of to be gunning for something is to try very hard to obtain something to be determined to get a job or to win a prize for example.

1. That's the second time Adam has gone behind my back and made up lies about me to the landlord. He's really gunning for me for some reason, I wish I knew why!
2. I'm gunning to be moved up a level in my class this week. I did really well on my last few tests.

1. Lucy has volunteered to take charge of all our recent projects, even though that has meant much longer working hours for her. It's obvious that she's gunning for a promotion.
2. I'm really gunning for the opportunity to represent the company at the town meeting this month.

1. As the market leader, every company we're in competition with is gunning for our top spot. We must do all we can to stay ahead of the game!