How to get a research professor position?
The first thing to realize is that within the same department a tenure track/tenured position is usually more desirable than a soft money position (what you are calling a research professor). Soft money positions tend to have less responsibilities (e.g., teaching and service) but at many research oriented universities TT staff can use grant money to buy out of teaching and service. In fact those on soft money positions often pick up the residual work thereby making their grant money go further.

Also realize that most departments do not want lots of soft money staff. There are often physical space constraints. Soft money staff can potentially push a department in a direction it is not interested in which can lead to social and political issues. There is also the issue of graduate students. Most people on soft money positions want to have access to graduate students but this can syphon off students and funds from TT staff and provide less security for the students (e.g., when the soft money runs out).

Finally, realize that most departments actually care about their staff. When soft money positions have a gap in funding, the members of a department often try and cover the shortfall. The ability to attract the best soft money candidates depends on the past history of how soft money staff have been treated.

Now to answer you question, at the highest I would contact the Department Chair. No Department Chair wants to hear from the Dean that they should consider someone. No Dean has the time to read CVs of candidates that are not good enough for a TT position. Even better is to contact a TT staff member in the department that you would like to work with. the best soft money positions are ones in which the individual is not isolated from the rest of the department.