Your cough may be a symptom of asthma--your other symptoms make this very likely. It is especially likely that your symptoms are from asthma if you have a history of eczema or other allergic rashes, if your symptoms come and go depending upon the season, if your symptoms began as an upper respiratory illness but persisted afterwards, or if you get worse with exposure to cold, chemical smells or fragrances.
You need to be evaluated by your doctor. Asthma symptoms, including cough, can be relieved with inhalers or other medication. In order to diagnose asthma, your doctor may want to do a breathing test to measure how well you move air through your airways. These tests are named "spirometry" tests. Sometimes asthma can be diagnosed by simply trying out treatment with an inhaler that is a usual treatment for asthma, to see if it relieves your cough symptoms.
One important trigger of asthma and cough is acid reflux from the stomach (known as gastroesophageal reflux disease, or GERD). If you have symptoms that suggest you are having reflux, then GERD may be another important thing to consider as a source of your symptoms.
Do you have a recurring sour taste in your mouth, or do you frequently notice burning pain (called heartburn) behind your breastbone (sternum)?