Like most damage over time effects, Serpent Sting cannot crit and thus does not benefit from any increases to crit chance you have. The two piece set bonus from the Windrunner's Pursuit/Battlegear set grants Serpent Sting ticks the ability to critically strike. Each tick then has a chance equal to your ranged crit chance of dealing double damage. The end result is that the damage of Serpent Sting is increased by your crit chance, so that if you have a 60% chance to crit, Serpent Sting will deal 60% more damage on average.
Efficiency (MarksmanshipTalent): Reduces the Mana cost of your Shots and Stings by 3% per rank (5 ranks total).
Hawk Eye (Survival Talent): Increases the range of your ranged weapon by 2 yards per rank (3 ranks total).
Improved Stings (Marksmanship Talent): Increases the damage done by your Serpent Sting and Wyvern Sting by 10%/20%/30% and the mana drained by your Viper Sting by 10%/20%/30%. In addition, reduces the chance your Sting damage over time effects will be dispelled by 10%/20%/30% (3 ranks total).
Noxious Stings (Survival Talent): Increases the damage done by the hunter by 3% on targets affected by their Serpent Sting.
Glyph of Serpent Sting increases the duration by 6 sec, from 15 to 21 sec, adding 2 ticks of damage. This adds 484 damage at rank 12.
The Tier 8 2 piece set bonus increases the damage dealt by 10%.
The Tier 9 2 piece set bonus allows the damage done by Serpent Sting to critically hit for 100% additional damage (+50% prior to patch 3.2.2).
Tips and tactics
Serpent Sting is a fairly weak damage over time spell, in comparison to those of other classes. To get the most of out it, it should be used in fights that will last longer than 15 seconds. On short fights and trash mobs, the hunter may find it is not an efficient use of mana. For Marksmanship hunters, it is worth using even in fairly short fights as it increases Chimera Shot's damage.
When fighting multiple enemies, it may be useful to apply Serpent Sting to all targets. If there is a large number of enemies, however, it is better to use Volley. Marksmanship hunters who have Glyph of Serpent Sting can use Chimera Shot to keep Serpent Sting up indefinitely on two targets, although this will require timing as the margins for error are very small.
Its greatest utility in PvP is against rogues and druids, to prevent them from entering stealth. (Druids can cleanse it, but will have to shift to caster form to do so.)
Be careful using this spell in situations where your group may use crowd control (CC), as the damage will break the effects. If you have used Serpent Sting on the wrong target, you can overwrite it quickly by casting Scorpid Sting on the same target, which does not deal damage and thus does not break CC.