Do I need exterior grade cables or conduit to run wire in an enclosed soffit?
While soffits usually have some venting in them, the interior of soffits are interior spaces not much different from unfinished attics to which they connect. Attics do not require different wiring from other interior spaces, so it doesn't seem that soffits should either. While attics (and basements) may often be more humid than some other spaces in a wiring scheme, these do not pose significant problems for normal interior cabling and outlet boxes.

The fixture illustrated has a weatherseal around the base (as do all exterior fixtures) to keep water from infiltrating. On the other side of that seal, no liquid water should be present.

It seems likely that the same applies to CAT5e cabling.  However, as has been discussed elsewhere, fire rating of low voltage cable is an issue. Use properly rated cable.