Deciding locants of double bonds vs. other substituents After having decided that the keto-group is the highest priority, this is a ketone. However, does the double bond become C2, or does the amino become C3?
In locant placement, do we prioritize double bonds over substituents, or other substituents over double bonds? (Or is it just a matter of having the smallest starting number?)

5-aminocyclohex-2-enone
3-aminocyclohex-5-enone
According to the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) while numbering the parent chain, an unsaturation is preferred over a substitutent.

P-14.4 Numbering
When several structural features appear in cyclic and acylic compounds, low locants are assigned  to them in the following decreasing order of seniority.
.. (a) fixed numbering of a ring or ring system;
.. (b) heteroatoms in chains and in rings and ring systems;
.. (c) indicated hydrogen;
.. (d) principal characteristic groups and free valences (suffixes);
.. (e) added hydrogen;
.. (f) saturation/unsaturation (hydro/dehydro prefixes and ‘ene’ and ‘yne’ endings);
.. (g) detachable alphabetized prefixes, all considered together in a series of increasing numerical  order;
.. (h) lowest locants for the substituent cited first as a prefix in the name;
..

As per the preference order the ketone will get the priority first as it is the principal characteristic group. Thus the name will end in an -one.
Now as the unsaturation is preferred over a substituent the numbering will be done in such a way that the double bond gets the lower number.
Thus the preferred IUPAC name of this molecule would be (5S)-5-aminocyclohex-2-en-1-one.