I found these 7 tips on completerunning.com about breathing while running:
1. Try slowing down before attempting to change your breathing.
2. Focus on longer, deeper breaths.
3. Breath more from your diaphragm (belly) than your chest (chest breathing is more shallow).
4. Focus on exhaling more fully. This will remove more CO2 (Carbon Dioxide) and allow for deeper inhalations of O2 (Oxygen).
5. Allow air to enter through both the nose and mouth. This will allow for maximum O2 intake.
6. Aim to take three footsteps for every inhale, and two footsteps for every exhale (3:2 ratio). A good descriptions of this can be found on Military.com:
Many experts will say that to fully oxygenate the muscles and clear the body of carbon dioxide you should breathe a 3:2 inhale-to-exhale ratio; full inhales and full exhales. This means you INHALE on the LEFT, RIGHT, LEFT foot strikes and EXHALE fully on the RIGHT, LEFT foot strikes. This pattern is not that hard to turn into a habit, but it may require you to slow your pace down for a few runs to master the technique. You will notice a lower heart rate as you are able to get more oxygen in and more importantly push all the carbon dioxide out of your body. You may notice that you naturally drop to a 2:1 ratio when you are really pushing it to the finish. That is OK. But realize it is difficult to maintain a pace that requires you to breathe at a 2:1 ratio.
7. One last thing – try not to over-think your breathing! It’s best to try to slow down, relax, and let yourself fall into your body’s natural rhythm.