The rules for numbering of bicyclic compounds such as azulene are given in Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book).
First, you need to generate a "preferred orientation" of the molecule:
P-18.104.22.168.2 Polycyclic fused ring systems are oriented in accordance with the following criteria considered in order until a decision is reached:
(a) maximum number of rings in a horizontal row;
Because of this, the two rings need to be horizontally aligned with each other in the preferred orientations. Note that this rule alone does not distinguish between the two orientations on the left.
Then, the numbering proceeds as follows:
P-22.214.171.124.1 The numbering of peripheral items in the preferred orientation starts from the uppermost ring. If there is more than one uppermost ring, the ring furthest to the right is chosen. Numbering starts from the nonfused atom most counterclockwise in the ring selected and proceeds in a clockwise direction around the system, including fusion heteroatoms but not fusion carbon atoms. Each fusion carbon atom is given the same number as the immediately preceding nonfusion skeletal atom, modified by a Roman letter 'a', 'b', 'c', 'd', etc.
This leads to two possible numbering systems for the two orientations above. Which one is preferred is given by a further rule
P-126.96.36.199.2 If alternative numberings for a ring system remain [...] the following criteria are applied in order until a decision is reached: [...]
(c) low locants are assigned to fusion carbon atoms;
The ketones in your molecule have greater seniority over the thiols (P-41), and hence the ketones are expressed as the suffix (leading to a name ending in "dione") instead of the thiols (a name ending in "dithiol"). Note that if the thiols are deprotonated this seniority ranking would be reversed, as anions have greater seniority over ketones (hence you would have a name ending in "bis(thiolate)").
The quinone derived from azulene is named as azulene-x,y-dione, where x and y are the locants of the two carbonyl carbons (P-188.8.131.52.3). The asymmetry in the quinone leads to ambiguity in the numbering, which is resolved by the next rule:
P-14.4 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);
Finally, the –SH groups are inserted as prefixes into the name, as they can no longer be suffixes. The appropriate prefix for –SH is 'sulfanyl'. The multiplier used is "bis" and not "di" to avoid ambiguity:
P-16.3.6 The prefixes 'bis', 'tris', 'tetrakis', etc. are also used to avoid ambiguity:
(a) before a mononuclear subset of a polynuclear acyclic structure;
Examples: bis(sulfanyl) (preferred prefix; defines two –SH groups, see P-63.1.5; whereas disulfanyl defines the –SSH group; see P-63.3.1)
As you can see, the manual generation of such names is in no way trivial. Most people use software to automatically generate names, which is generally sufficiently clear for the chemical community to understand them. However, it should be mentioned that these automatically generated names are not always the preferred IUPAC names: for example, ChemDraw 16 gives "2,6-dimercaptoazulene-1,5-dione" as the name, even though the prefix "mercapto" is no longer recommended (P-63.1.5).