The key to increased competitiveness is a willingness (or proclivity) to take things personally.
Many competitive people are personally offended when they fall behind, are criticized or outperformed.
Competitiveness is instinctual. I compete in so many things it becomes embarrassing to mention. I don’t know how to coach it, but I would imagine you need to find a way to personally invest in what you are doing more than you are right now. If you are a casual participant, keeping your emotions cool, it’s unlikely your competitive spirit will ever be triggered.
Competitiveness is emotionally linked to the fear of failure. If you can teach yourself to panic on command, and visualize the negative consequences of you losing, you would be able to drive yourself into finding a will to win in order to avoid those negative projections.
It may also be helpful to identify a rival. Who else is going for exactly what you are going for? Put a face to the person who is trying to limit your opportunity. It always takes at least two to compete. Sometimes focusing on a counterforce can be helpful in rousing the competitive spirit.