How can I help bring up mucous when I cough?

Mucolytics. Mucolytics are medications which "thin" mucus and make it easier to expectorate (cough up). Guaifenesin is the classic example of such a medication. It is available in a short-acting syrup (robutussin) and a long-acting tablet (mucinex). Additionally, when there are other pulmonary conditions such as asthma and copd, medications which help treat these conditions will help in bringing up mucus.
Increase Fluids. It depends on the cause of the cough but i would recommend increasing fluids. However, i would avoid dairy which seems to increase mucous production. And i would also use a guiafenesin type cough syrup which helps break up mucous. Finally, if it is a dry inflammatory cough you may need to use an anti-inflammatory inhaler or medicine as in asthma or copd.
Fluids... Increasing your fluid intake will help keep your secretions thin enough to cough up easily. A non-productive cough could be a sign of other diagnoses like bronchospasm, gerd, post-nasal drip, etc. So see your doctor for any persistent, non-productive cough.