Naming a hexasubstituted benzene ring
The most important simplified criteria for the numbering are:

lower locants for the group that is expressed as suffix
lower locants for multiple bonds
lower locants for prefixes
lower locants for substituents cited first as a prefix in the name

The corresponding actual wording in the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) reads as follows:

P-14.4 NUMBERING
When several structural features appear in cyclic and acyclic compounds, low locants are assigned to them in the following decreasing order of seniority:
(…)
(c) principal characteristic groups and free valences (suffixes);
(…)
(e) saturation/unsaturation:
  (i) low locants are given to hydro/dehydro prefixes (…) and ‘ene’ and ‘yne’ endings;
  (ii) low locants are given first to multiple bonds as a set and then to double bonds (…);
(f) detachable alphabetized prefixes, all considered together in a series of increasing numerical order;
(g) lowest locants for the substituent cited first as a prefix in the name;
(…)

The compound that is given has no principal characteristic group that would be expressed as a suffix. Thus, Rule (c) is not relevant in this case.
No locants are assigned to the delocalized electrons of the benzene ring. Thus, also Rule (e) is not relevant.
Usually, Rule (f) is important for benzene rings with multiple substituents. In this case, however, all six substitutable positions are completely substituted; i.e. the locant set (all considered together in a series of increasing numerical order) is always ‘1,2,3,4,5,6’ – no matter how we turn it.
Finally, Rule (g) stipulates that the lowest locant is assigned to the substituent that is cited first as a prefix in the name. Therefore, the correct name for the compound that is given in the question is 1,3-dibromo-5-chloro-2-ethyl-4,6-dimethylbenzene.