Blitzcrank is:

+ Very fun
- Hard to master
+ One of the best supports in game
- Team dependent
+ Hybrid role
- Item reliant

PROS:

- Most CC chain champion in game (knock up, pull, silence, knock up)
- Quite high AD scaling in the game.
- Versatility (Can play AD, AP or Support)

CONS:

- If you can not land crucial pulls, Blitz is useless. Skill shot based.
- Counter picked by long range poking champs.
- Can accidentally pull wrong champ and ruin team fights.

Flash
Great choice for Blitz, as he can engage team fights with a flash ulti, or a flash grab to secure a kill.
Ghost
Ghost is good for both offensive and defensive. Can be used to gain distance and Powerfist or used to flee a fight.
Heal
Heal is always good, depending on your play style. Early game aggressiveness is key to have heal.

Greater Quintessence of Desolation x3

- Armor pen helps with early/late game damage.

Greater Mark of Desolation x9

- More Armor pen to up his relatively low damage will playing support/tank.

Greater Seal of Resilience x 9

- Armor to help him damage sustain in lane early game.

Greater Glyph of Warding x 3

- Minor Magic resist to help with stain early AP damage.

Greater Glyph of Shielding x6

- Magic res per level to help your MR scale into late game.

The points taken in the mastery tree is 9/12/9. A quite unique build.
It covers all Offence, Defense and Utility.

Offense
The Offense tree covers minimal physical damage, attack speed and armor pen. For the upper edge in lane early game.

Defense
The Defense tree covers basic magic res, armor, health regen per 5 and flat health. This will help you with held some early game damage.

Utility
The Utility tree covers mana regen per level, low movement speed, flat mana regen and extended buff duration. Which is always good.
The Ul

Starting items:
- Faerie Charm, 3 Sight wards and 3 health pots.
or
- Faerie Charm, 1 Vision ward, 2 Sight wards and 1 health pot.

Core Items:
- Philo stone
- Heart of gold
- Boots 1
- Oracles

Final Build:

- Shurelya's Reverie
- Aegis of the Legion
- Frozen Heart
- Banshee's Veil