If you and/or the person concerned is a regular worshipper, the clergy will appreciate being kept in touch with the process. Most of all, they will want their prayers for you all to be well informed and up to date, so a weekly email (perhaps more frequent towards the end) will be much appreciated.
Secondly, they will want to know whether you or the patient would welcome a visit. In some traditions, this could include one form or another of last rites - eg Communion or anointing. Most clergy are very pleased to be given a chance to exercise this part of their ministry, so there is no need to feel shy or coy about asking for it. And don’t worry about the logistics: they will bring what, if anything, they need.
It is possible - even probable - that a minister who knows you well will encourage you to discuss with the patient what form they would like their funeral to take. Have they favourite hymns, prayers, readings? The Minister might raise the subject directly with the patient, but s/he will be glad to have your help, not least because most people need time to reflect on such questions.
Clergy are busy people and are likely to be responding to a number of requests for prayer, visits, time at any given moment. So the more notice you can give and the more ready you are to accommodate their schedule, the easier the relationship will be. (If s/he’s late, it is not because they forgot or fell asleep. It is because they had an urgent call to the hospital or the prison. So be forbearing.)
You may want to think whether you would like the minister to be present at the moment of death. This used to be quite normal, but is now much less so. If that would mean a lot to the patient, raise it with the minister. Most will agree readily in principle, but it will always have to be dependent on competing claims on their time.
And if the patient has little or nothing to do with any religious set up? Then there is no need to be in touch with the clergy, at least until after the death. Then the undertaker will take charge of negotiations about the funeral and there is no need for your involvement. However, some clergy express the wish to visit the bereaved family, partly to discuss the arrangements for the funeral but more to get to know you as people and to learn what they can about the deceased. This makes it easier for them to make the funeral special for the family as a whole. You are likely, therefore, to find it time well spent.