The Shulchan Aruch lists a series of activities after which one should wash one's hands. They include getting up in the morning, having marital relations, visiting a cemetery or other contact with a corpse, coming out of a restroom or bathhouse, and removing shoes or touching certain parts of the body such as the head or the feet. Washing upon waking in the morning should be done three times alternating hands with a cup. The other activities require neither three times nor a cup, although some have the custom to do so anyway, particularly after visiting a cemetery or having marital relations.
Some of these activities, such as washing hair and removing shoes, take place while preparing for the mikveh. However, use of the mikveh would subsume the need for hand washing. In the process of getting dressed it is possible that one touched one's feet.
Another issue is that the mikveh is a form of bathhouse. For this reason, many people practice hand washing – once on each hand, right and then left, without a bracha – when they leave the mikveh. Others hold that, as bathhouses are far cleaner today than they were at the time of the Shulchan Aruch, one is not obligated to wash hands. Thus, a person who generally washes with a cup on leaving the bathroom should do so on leaving the mikveh. However, someone who just washes with water from the faucet when leaving the bathroom is not obligated to do more than that when leaving the mikveh, although there certainly is no problem with doing so.