Should I send private email to department's dean after sending faculty job application?
If you really know him and have corresponded otherwise, then it would be fine. Otherwise I wonder, as would everyone else, what you are trying to accomplish. 

But you haven't said that you have actually applied. The title is a bit inconsistent with the first paragraph. 

However, if you get an email, after applying, thanking you for the application, you could reply to that, reminding him of your earlier connection. But it would probably be improper of him to take any action other than what he would do for any candidate. Likewise, if you get an interview, you could go visit him. 

But probably not a cold email. Especially since you raise it as a potential issue.