No, you cannot.
You quoted the relevant rules (emphasis mine):
When you hit a creature with a weapon attack, you can expend a charge to deal an extra 4d6 necrotic damage to the target, and you regain a number of hit points equal to the necrotic damage dealt.
The rule says that you can spend a charge (only one), and does not provide further specifications about spending more charges for increasing the damage.
It is true that if you have the Extra Attack feature, and hence you can attack more than one time, for each attack that hits you can spend a charge for activating the tattoo feature. For example, if you can attack twice and you hit both times, you can deal 4d6 + 4d6 necrotic damage (one for each hit) and you can regain the same amount of HPs.
Some items allows to use more than one charge: the effect of using multiple charges is in the item's description. For example, the Alchemical Compendium reports (emphasis mine):
As an action, you can touch a nonmagical object that isn’t being worn or carried and spend a number of charges to transform the target into another object. For 1 charge, the object can be no larger than 1 foot on a side. You can spend additional charges to increase the maximum dimensions by 2 feet per charge. The new object must have a gold value equal to or less than the original.