The paragraph starts with the choice of an article. A definite article is required since we are talking of a specific man. Tall, dark, good to look at are qualifying the man. To keep the parallel structure (similar things to appear in the same part of speech) intact they have to be in the same format. Thus we write good looking or handsome instead of good to look at. And should separate the last two adjectives. Sight (ability to see) should be replaced by site (place of incident).Had visited is the wrong tense. Since it is the second past action past perfect tense is not needed. Past perfect is used only for the first action in the past when two things take place. Departed is used for people who have died. Moreover had left is needed as it is the first past action. The modifying clause should be close to its subject so wearing a long blue overcoat has to be placed near man and not detectives. Were is a plural verb but man is singular so use to was. Which is used for non living things and animals so change to who. For is used for an indefinite span of time, since for a point of time and from for a range of time. Here since is an apt choice of preposition. It should be serial murders and not murder as it is a chain of murders. Was should be changed to were and the past continuous tense should be changed to simple past tense as the action is complete. Use ago as it refers to a fixed period backwards, before is indefinite.
Mukta Mehta