The fighter's Action Surge feature (post errata mentioned in the 2019 Sage Advice Compendium) now states:

On your turn, you can take one additional action.

The Eldritch Knight fighter gets the Arcane Charge feature (PHB, p. 75), which states:

At 15th level, you gain the ability to teleport up to 30 feet to an unoccupied space you can see when you use your Action Surge. You can teleport before or after the additional action.

Are you allowed to take Action Surge's "additional" action, thus allowing you to use Arcane Charge, and then take your regular action?

If you have to use your normal action first, then your options are Action --> Teleport --> Action Surge and Action --> Action Surge --> Teleport. If you can use Action Surge first, this gives you the new option of Teleport --> Action Surge --> Action.

Yes, you can Action Surge before taking your first action.

In the PHB, Action Surge doesn't mention a condition of using your Action Surge aside from being on your turn. If it had said, "after using your action" this wouldn't work, but because it has no such specifier, you can use your Action Surge before your original action.

Strict RAW: The "Action Surge action" is always your second action

The only mechanical description of how Action Surge works on the turn it is used is:

On your turn, you can take one additional action.

As written, Action Surge isn't a feature that you "activate" to gain an additional action, which you can then use at any time later in your turn.^* You use your Action Surge feature simply by taking an additional action, and you can't take an addition action until you've already taken an action first. If you haven't begun taking a second action on your turn, you haven't used your Action Surge (yet). Hence, there is no way to use Arcane Charge before taking your first action, because the earliest you can use it is right before you start taking your second action.

To put it another way, when you use Action Surge to take an additional action, while it is often useful to speak in terms of a "regular action" and an "Action Surge action", you really just have 2 identical actions, and the only difference between them is that one happens after the other. Action Surge is only used if you actually take that second action, after which the second action becomes unavailable until your next rest.

A workaround: Ready your first action, then teleport

There is a way to use your first action after the teleport: by readying it. For example, you could use your first action to ready an attack against your target as soon as they are within range, then, then starting your second action and teleporting to the enemy. Your readied action triggers, granting you 3 attacks, and then you make 3 more attacks with your second action. (Note: I'm assuming a 15th level fighter, with 2 Extra Attacks, since Arcane Charge is gained at 15th level.) The only "cost" to doing this is that you must use your reaction to trigger your readied attack action, which leaves you without the ability to make an attack of opportunity until your next turn. Of course, this also works if one or both of your actions isn't attacking with a weapon. For example, you could deliver 2 touch spells to your target by readying the first one and then teleporting and releasing it as a reaction, then casting the second one.

Rules as fun: Go ahead and allow it

The above argument is a pretty technical, and there's really no reason as DM that you can't rule that as soon as the fighter decides that they want to use their Action Surge, they gain the teleport to use before or after either action. This provides a very minor situational increase in the fighter's power: specifically, they will be able to make their full 6 attacks against an enemy that wasn't within their movement range by teleporting to them before their first action, and have their reaction left for an attack of opportunity.

The one thing to be careful about in this case is when the fighter uses their Arcane Charge teleport, then takes one action, and is then prevented from taking their second action (e.g. by being knocked unconscious). If this happens, you should consider the Action Surge as having been used as soon as the teleport happens, because even though the fighter didn't take their second action, they used a feature that depended on them using the Action Surge. If you don't rule it this way, there may be some exploits that allow the fighter to gain infinite teleports by repeatedly teleporting and then "cancelling" their Action Surge by intentionally incapacitating themself.

^*For an example of something that does work in an "activate now, use later" way, see the Dash action, which says:

When you take the Dash action, you gain extra movement for the current turn.

If Action Surge was meant to work like this, it should say something along the lines of:

When you use your Action Surge, you gain an additional action that you can use during the current turn.