If understand the halacha correctly, one must do netillat yadayim after touching parts of one’s body that are normally covered (by clothing). I work as a doctor and almost every patient I see, I examine. This invariably involves touching the other person’s skin that is normally under clothing - and even sometimes, body parts considered unclean by the wider society. My normal practice is to wash my hands with water from the tap plus soap and/or alcohol gel. Is this sufficient or should I wash with a kli? If so, should I do so after each patient or just before engaging in non-medical activity (e.g. lunch, mincha, etc.)?

Washing your hand with water covers the obligation of Netilah for these issues. [Sheilat Shlomo 7: 9]