When you delete your account all of the personal information contained there gets set to be completely removed within 30 days. Any links you created in that account will continue to work and direct to the most recent destination it was set to land on. Your bit.ly links will not break.
You will no longer be able to track metrics for your links, edit them or redirect them. If your links use a custom domain those links will continue to work as long as the domain you're using is still directing to Bitly's IP addresses in their DNS settings.