Because his behaviour is unfair, it's terribly tempting to try to get your own back by pointing out his shortcomings to others, especially your boss. But this is likely to be seen as bitchiness, reflecting badly on you, rather than your rival. Turning your attention from your competitor to thinking about how you want to behave is the key to stopping this unhealthy triangle you're locked into. Of course, it's unwise to switch off from his machinations completely - you need to watch your back - but don't waste time and energy developing paranoid delusions about him.
Concentrate on doing a good job, particularly where your direct reports are concerned. If you have 360 degs appraisals, chances are your feedback and scores from your people will be more positive than your rival's. See what other external measures will testify to your achievements and qualities. If there is an internal newsletter, use it to broadcast news of your team's wins.
If you're hoping to head another department, find out what's going on there, and forge relationships with any members of senior management who have a role in making appointments.