I'm inclined to agree with FoxyBoxes after some playtesting. This should be 3rd level at least. It's strictly better than Misty Step while being a level lower and grants advantage and a free attack on up to 3 creatures if they're lined up.

Definitely needs to be a higher level.

Target an empty space within the spell's range that you can see.  Move five feet directly toward the chosen location.  During this movement, you do not provoke opportunity attacks.  In addition, ignore any obstacles or creatures blocking the path as if you were ethereal.  After moving, you may make a single melee attack with advantage against a creature within 5 feet of you that you have not attacked as a result of this spell.  After resolving the attack, repeat this process by moving another five feet and making an attack until you reach the chosen location.

Misty Step is a Bonus Action, this isn't. It requires an Action.

That said, it allows you to basically roll a Misty Step-like effect into an attack action, which is sorta like having used your bonus action but without actually using it... but only if you had planned on making an attack... with a melee weapon. So the loss of range, with similar utility or mildly superior utility, means this is extremely comparable to the same exact power level as Misty Step itself... which is a 2nd level spell.

Anyone casting this instead of any 3rd level attack spell is clearly making an error in judgement. As any/all of them are superior. Just take a look at Thunder Step for a similar analog of what a 3rd level damaging teleport does. It is superior to this in every way. So 3rd level this spell is not.

I think the only other issue I have with it is that it is a bit weird that this is an evocation spell.

I love the spell concept.

If you want to stick with the established literature most teleportation spells are from the school of conjuration. 
In my game world rather than make this spell a teleportation I rule that the caster just moves with incredible speed by transmuting and altering the the spatial and temporal dimensions of the environment.

If I'm being completely honest, it's only evocation because I was using it with an Evocation Wizard/Fighter Gish when I made it, and I wanted it to be maximize-able and get the Int damage bonus.

Spell schools have always seemed entirely arbitrary to me and seem to be a factor of balancing as much as they are flavor. I saw Eldritch Knight and Arcane Trickster only getting certain spell schools and thought "Because if they got this, that, and the other, they'd be much stronger." To test that, I unlimited those classes spell choices (still within the wizard list) in all my games. Lo and behold, they outperform other subclasses before level 5 by a wide margin until the release of Rune Knight. That's of course only personal experience, so I can't really call it anything conclusive. Anyway, that's why it's Evocation.

I also just don't really know what each of the spell schools represents outside of their most powerful effects being what I equate them with.

anime quickdraw attack