For a professor in the US, How important is one's relationship with the dean?

The question, although reasonable, seems hard to answer in general terms. Programmatic change is generally not done by individual faculty members in a department, but by groups and/or committees of them, in confluence with the department Chair/Head.

Many programmatic changes do not "go through" the Dean -- in particular "day to day" business usually does not -- but those involving nontrivial financial changes usually do. For all but the smallest colleges it is rare for individual faculty members other than the Head to have substantive conversations with the Dean about such things: again, it would be better if they went through the department as a whole and/or the Head. When it comes to financial outlays that must be approved by the Dean: I've seen plenty of instances where the Dean approves but "just doesn't have the money," so I would have to think that even mild disapproval of the initiative would sink it pretty thoroughly.

Perhaps the most useful thing I can say is: look at where the program is now and look at where it was in the recent past. If you are really unhappy with the current position or the perceived trajectory of the department, then you should not count on being happy with it in the future (no matter who the Dean is). Conversely, if those look reasonably good to you, talk with the faculty and see how they feel about the future. If there's some important change coming around the bend, I hope they will let you know. But it is more likely that being in the department for the next ten years or so will be roughly like having been in the department for the last ten years or so. The Dean does not reside in the department, so unless s/he is actively at war with the members and interests of the department, some disagreement in the overall vision need not affect the actual working life of a professor in the department unduly.

By the way, the bit about the Dean "may leave in the next 5 years" sounds way too vague to take seriously. Maybe s/he will and maybe s/he won't. If s/he does, maybe the next Dean will be better or maybe worse. It is very hard to plan for the future with regard to this kind of thing: honestly, I probably wouldn't even try.