Why do my legs hurt when I drink alcohol? Sometimes when I have a drink of alcohol, I get a severe ache in my legs that lasts about 3-6 minutes.

No clue. Some people are very sensitive to the effects of alcohol on muscle, where it can cause inflammation of muscle cells (myositis), but this usually occurs after binges of large amounts of alcohol, not one drink. There are always idiosyncracies where people are more sensitive. Having a blood test that can show normal or elevated muscle enzymes can be ordered by your doctor, who you should talk to.
You . You may be dealing with something as simple as dehydration and/or vitamin deficiency, most likely potassium. I recommend you stay hydrated(with water or gatorade), especially before drinking alcohol. Also try taking a daily multivitamin. When the ache occurs try eating a banana or drink some coconut water, both are high in potassium, and may help alleviate your symptoms. You may also consider scheduling your annual history and physical with your doctor, in order to assure you are not suffering from any underlying illnesses.