The rules in SC book say that infected character can use Essence Drain ability to fuel their Essence up to twice their natural maximum scoring ( above this, each point of Essence is lost ). In the same page, you can find that the infected character start with 5 Essence and 1 Magic. But how is the max Essence? Is it the initial 5 or 6 ( like othe character )?

Maximum natural Essence is 6. Infected with Essence Drain may obtain up to 12. This is RAW.

Interaction with augmentation, however, is unclear.
Does augmentation reduce current Essence or Natural Maximum?
If the former, Essence loss from augmentation basically does not phase the Essence Drain Infected at all, which leads me to believe it is incorrect.
If the latter, then augmentation hurts Essence Drain Infected significantly, at least as much as mundane characters, it significantly hampers one of their key abilities.

That being said, the book seems to imply the former is correct, but is not at all clear. My suggestion is as follows (I think it is a House-Rule):
You have your Essence as normal. Augmentation reduces your augmented Essence. Essence Drain can increase your Essence up to twice the Natural Maximum (6). This is not considered an augmentation, but an increase to the actual Essence score - the Essence Loss weakness works the same way.

Thus, a normal character has 6 Essence. The same character with 0.6 points of augmentation would have 6 (5.4) Essence.

A Vampire would have 5 Essence. The same character with 0.6 points of augmentation would have 5 (4.4) Essence.
If said vampire where to drain 3 points of Essence, it's Essence would then be 8, or 8 (7.4) with augmentation.

Jesus wept.

Okay, let's go through p.77 on Runner's Companion.

A normal metahuman's natural Essence (and thus natural maximum Essence) is 6.

Characters starting as Infected begin with an Essence of 5. Their natural maximum Essence is still 6, for whatever that's worth. If the Infected character doesn't have Essence Drain or Essence Loss, then any further Essence loss they suffer (from Essence Drain, addiction, augmentation, etc.) is handled normally.

If the Infected does have Essence Drain/Essence Loss, things are handled a little differently. Their natural maximum Essence begins at 6, and using Essence Drain they can increase their current Essence up to twice that (12). Any Essence loss not caused by the Essence Loss weakness - such as addictions, or augmentation - are subtracted from their natural maximum Essence. Reductions in their natural maximum Essence can also reduce the maximum Essence they can retain with Essence Drain.

So, just for shits and giggles, let's take a starting vampire with Essence 5. His natural maximum Essence is 6, and with his Essence Drain power he can bring his current Essence up to 12. If he takes 1.0 Essence worth of cyberware and bioware augmentations (after all calculations are complete) at character creation, it is subtracted from his natural maximum Essence (which is reduced to 5), and his maximum Essence is 10. His current Essence (5) is less than his maximum Essence (10), so he doesn't lose any Essence for getting the implants.

To make it simple.
A vampire can get upto 5 Essence points worth of cyberware.
A vampire max Essence is current natural (if not a vampier) Essence(round down) times 2.
So a Vampire with 1.1 Essence worth cyberware/bioware now has max of Essence of 8

Natural Essence 6 - 1.1 Essence of Cyberware/bioware = 4.9 or 4(rounded down)
and a Max Essence of, 4 x 2 = 8 max Essence