Fastest, and most efficient way to build a golem

[RoboEmperor]

What class would be the best option?

Artificer

What feats are required?

How soon could this happen?

Craft Construct at level 6. Legendary Artisan, Extraordinary Artisan, Exceptional Artisan, and Magical Artisan reduce the cost to create quite a bit.

What does it cost to create a golem?

How realistic is this goal?

Hydra Effigies are playable. They're really cost efficient.

If you just want constructs there are other methods.
1. Cast Personal Weapon Augmentation on a Colossal Weapon to give it the Flying property (magic of faerun) to turn it into a Colossal Animated Object for 10min/level starting at level 1. If you go Illumian and take a level in Cleric you can use Divine Metamagic:Persistent Spell to persist it instead at level 2.
2. As an outsider wizard or sorcerer grab the feat Assume Supernatural Ability:Animate Objects, Alter Self into a Ravid, and use its CL20 Animate Objects (Su) ability to perpetually animate a Gargantuan object. Achievable at level 3.
3. As an Artificer at level 6 craft scrolls of Simulacrum and create Simulacra of Golems. You're using the Divine Crusader Spell List to create CL6 Scrolls of Simulacrum.

It is even earlier if you use lesser planar binding for getting a mirror mephit to cast simulacrum for you.

Improved Familiar is faster.