Is heart attack chest pain typically a deep feeling in the chest or closer to the surface?

Usually deep: But there is not an answer that fits every occurrence. Cardiac chest pain is usually dull ache or squeezing sensation beneath the breastbone, sometimes with radiation to the jaw, shoulder or arm, usually the left--most often with exercise or stress.

Can be either: Chest pain with a heart attack comes from the damaged tissue causing irritation of the tissue around it. Shoulder and arm pain are from irritation of the diaphragm, chest pain from irritation of the chest wall, etc. The location of the pain depends on the location of the inflammation.

Dull: Typical pain is a pressure or heaviness, tightness sensation. However patient's perception of pain is different.