Hello I am NuclearNoobian and welcome to my first guide! I got inspiration for making this guide ever since I saw the champion spotlight. I hope you enjoy the rest of the build and feel free to leave any comments below!.

I run Magic Penetration Marks Reds, Scaling Health Yellows, Scaling Cooldown Reduction Blues, and Ability Power Purples. Instead of running the standard Scaling Ability Power Glyphs on Ekko I bring along Scaling Cooldown Reduction because the 5% Cooldown from Masteries and the 20% Cooldown from Nashor's Tooth give you just barely 40% at level 18.

sneakyawwlewewewewewewewewewewe Starting Items
Not much to explain here only that take Crystalline Flask for when the enemy laner is heavy poke.

sneakysneakysneakysneakysneakysneaky!Core Items
Nashor's Tooth is very important to Ekko's build and must be gotten first. It allows Ekko to attack at a decent speed and offers him 60 AP. Even though this item doesn't immediately make Ekko's power skyrocket it is crucial he is able to get his passive off since it is his second most damaging ability. Also, the passive on Nashor's Tooth gives 15 (+15% AP) on hit, making up for the fact he didn't start with something like Luden's Tempest.
Luden's Echo follows Nashor's Tooth, giving him 120 AP. This item is built before Rabadon's Deathcap because of the 100 (+15% AP) passive and the 7% movement speed which can allow him to catch up to enemies and get his Z-Drive Resonance passive off.
Sorcerer's Shoes is standard for AP Midlaners to run and is what I would recommend almost all the time. It is possible to go for Swiftness Boots or Berserker's Greaves but is probably not the most optimal.

sneakysnneakysnweeasneakysneaky!Deathcap + Voidstaff
Voidstaff can be swapped for something different but generally you will want to include Voidstaff in your build path. Go with Rabadon's Deathcap first if you want a lot of AP and the enemy team doesn't have much Magic Resist and follow with Voidstaff. Go Voidstaff then Deathcap if they do have a lot of Magic Resist but remember your mini-powerspike will be delayed by going this route.

weweweweweeeweewewewewwewewewewew6th Item
In most situations I'd probably end with Lich Bane, Rylai's Crystal Scepter, or Zhonya's Hourglass. Lich Bane gives you a lot of damage after using an ability and autoattacking, Rylai's gives you 100 AP and can slow the enemy if proccing Z-Drive Resonance is a problem, and Zhonya's Hourglass is for defensive purposes. Zhonya's is actually more useful than you'd think, besides the defensive active if your ultimate is up then your clone will be 1.5 seconds "in the past." If the enemy team is surrounding you waiting for the active to end you can ult and deal the 130% AP ratio to the enemies around you. Them clustering around you basically sets up your ultimate. You can use this as a way to turn around teamfights or a very risky way to engage.

werwewwleweweweweweweweweeqweqweOddball Items
Blade of the Ruined King may actually serve some good purpose on Ekko. He can use the attack speed to proc his passive faster and use the active to deal Max. % Health damage. The active combined with the Parallel Convergence passive combined with the executioner mastery gives you quote, "tons of damage" - Phreak. Guinsoo's Rageblade is actually really good on Ekko, despite the low AP ratios. It's passive gives him 32% Attack Speed and 32 AP at max stacks and when below 50% health he gets great combat stacks that will allow him to sustain and melt faster.

I run 21/9/0 on Ekko, although some people may find use in biscuits from 21/0/9. I take a point in Spell Weaving since Ekko can blink to someone and auto to get the stacks needed. Take the points in Executioner since your Parallel Convergence's passive also pairs with it.

+ Gap Closers
+ Reverse damage from all-ins.
+ Reposition and restart fights with Chronobreak. + Vulnerable to CC
+ Chronobreak's AOE is hard to hit.
+ Parallel Convergence takes a long time to set up.
+ Chronobreak requires a setup for trades and all-ins.

Z-Drive Resonance
At 3 stacks of Resonance (Resonance stacks are gained by abilities and autoattacks) Ekko deals 15 + (12 x Ekko's Level) (+70% AP) in magic damage and slows them for 40/50/60/70/80% for 2/2.5/3 seconds. If the Resonance stacks proc on a champion, Ekko receives movement speed equal to the slows percentage and duration. Ekko's Z-Drive Resonance can be easily procced from his abilities. His Q, Timewinder, gives a total of two Resonance stacks if Timewinder hits the target twice. It is important his Z-Drive Resonance is procced when going in for any trades or all-ins. Try to proc his passive after you use your W, Parallel Convergence, to have a better chance of stunning the enemy.

I max Timewinder first because it is his only basic ability that is ranged and does decent damage. It slows to aid in escapes or all-ins. I max his E, Phase Dive, over maxing Parallel Convergence because its cooldown gets down to 5.75 seconds with the purchase of Nashor's Tooth and allows him to basically dodge one ability and blink away from another. I max his Parallel Convergence last because although the shield strength increases the stun nor the shield duration increases. And of course, make sure to put points into Chronobreak at every opportunity.

Since the release of this build (and before it) I've gotten questions as to why I rush Nashor's Tooth first. Understandably, it's an odd item that is somewhat rare to see built. Most people probably feel rushing Luden's Tempest or Lich Bane is a much better way to build him. I build it first because like most champions, you can't really one-shot people with your combo. Since Ekko's all-in combo requires him to be autoattack at least a few times, he needs Nashor's Tooth. It is much easier to deal damage to an enemy with a solid 1.0 attack speed compared to a 0.67 attack speed. It also grants him cooldown reduction which helps him trade more and more often. It's also a decent early item compared to Luden's because he doesn't have to save a full 1600 Gold to get a Needlessly Large Rod. Also, the passive on Nashor's makes the item actually be a lot more useful than 60 AP and +50% Attack Speed.
TL;DR: Ekko needs to combo his abilities and autoattacks in trading and all-ins which Nashor's Tooth helps with its attack speed, passive, and cooldown reduction.

Ekko's ultimate Chronobreak, has many unique properties (that are super op). His ultimate has a super huge AP ratio of (+130%) and heals himself for a percent of the damage taken during the past four seconds. Why his ultimate is so good is because he can teleport back to his recent path and deal AOE damage. Since he can teleport back to himself it has a somewhat broken property. Ekko can recall to fountain and buy items and instantly teleport back to his previous location. This can be used in lane to increase pressure because you basically never left the lane. Even worse, if he has homeguards, he can heal himself incredibly fast and teleport back with a new items AND the homeguard effect.