Cluster, migraine and tension-type headaches may produce a headache that awakens an individual in the early morning hours (usually after 4 AM), or is present upon awakening and is most intensive during morning.
When the one sided headache is constantly located at the same side, should prompt medical attention for finding a cause (brain tumor), cause unilateral migraine usually changes the affected sides of the head.
If this headache is starting at the back of the head, spreading upwards, to the side of the head and the forehead, it could be an occipital neuralgia.
Dizziness can be accompanied symptoms of migraines, occipital neuralgia, tension type of headache (tension headache can be unilateral, too).
Suffering from anxiety, depression, high blood pressure or having sleep apnea, makes one prone to morning headaches.
If you notice that the intensity of the headache is getting worse overtime, and you start to experience new symptoms, ask your physician for a brain MRI.

Best wishes!