This is a UNIQUE caster build for Ezreal. You can go tear and get muramana if you're not good with mana management but if you can play around it, it works better. This build does enough damage that you don't need to use too much mana to kill someone. I have dueled Ezreal's biggest counter Draven with this build and won! It has huge sustain, survivality, and damage.

max out Q for damage, poke, cdr, passive, and interactions with core items.

Max out W second because it only scales with ap so you want the base damage up. This is one of the biggest differences from normal Ezreal builds.

E scales with AD so even with just 1 point it gains damage from items. You lose positioning power but gain damage. going iceborn makes this route safe.

This is a full on damage masteries! recommended for the best Ezreal's out there only.
gives early game trading, roaming/pick potential.
For a safer route I would recommend two options swapping out TLD for Warlord's
ferocity: feast + warlord's + battering blow
Cunning: meditation + secret stash + dangerous game
or
Cunning: meditation + secret stash + dangerous game
Resolve: recovery + explorer + runic armor + perseverance

I go with a mix of AS+AD. I do this for farming.
Mana Regen for poking champs, last hitting minions when needed, and skipping tear.

Find what you're missing. I personally don't get any armor/mr on Ezreal.
You can get :
AD (best for any route)
Life Steal for survivability (best for safe route)
AS for damage+farming (best for safe route)
Mana Regeneration (best for spamming abilities)
CDR (best for spamming abilities)

I get AS+AD to farm.
Ezreal has a slow AA animation so the AS helps with that.
The mana regen also allows you to last hit some of those minions you might not be able to get while under tower, it's far, or two minions are dying at the same time.
Ezreal has a mid game damage spike so cull works well and the AS allows you to take advantage of the life steal cull passive.

1. Death's Dance:
it uses death's dance for damage, cdr, sustain, and survivability.
it gives the most AD in the game alongside blood thirster, since this is a caster build flat AD is crucial.
The CDR is also a must because of the same reason. caster builds need cdr to use their skills more often.
death's dance passive of taking 30% of the damage as a bleed for 5 seconds allows ez to recuperate his health if he gets caught. This is because Ezreal's E allows him to distance himself before he takes the full combo and then heal up with his Q's using the other passive.
Now, I understand that his Q counts as an auto allowing him to heal up with life steal but the reason for it is its synergy with iceborne.

2. Iceborn Gauntlet:
It gives armor for some tankiness against ad assassins and allows him to duel adcs better with the sustain from death's dance.
20% cdr because it's a caster build.
500 mana because it's a caster build, mana is required for skills.
The slow it provides allows ezreal to kite even better than he already can and allows you to support your team with the aoe slow.
Now, here comes the biggest reason for this item choice...
Its synergy with death's dance.
Every champion/minion/monster it hits he will heal up from the iceborn aoe.

Sterak's Gage (vs. assassins or if behind):
It gives 30% base damage witch essentially is a small buff to his AD for AA's and his AD base skills but also a small buff to iceborn's spellblade passive. let's say his base damage is 80. It would normally do a total of 80+80=160 damage with spellblade, however, with Gage it will now do 104+104=208, 48 AD.
It gives 400 health for more sustain.
The passive Sterak's fury is where it shines. With this item he actually strives against some AD assassins where a lot of ADCs can't compete. After their burst and their abilities are on CD ezreal will gain a burst of strength to his Q's for 8 seconds. using the same example of 80 base damage, Gage will give 136+136= 272 damage! 112 more damage from a 2600 gold. It also gives a decaying shield which would take the 30% damage bleed from his death's dance and give him time to heal up with his burst of damage.

4. Duskblade of Draktharr. (preferred vs squishies or when ahead for snowball):
It gives 55 AD for his skills since it's a caster build.
10% CDR benefits him because he is a caster build.
+18 lethality gives him a boost against squishies such as other adcs and most mid laners.
Nightstalker passive gives him a huge burst of damage, going from 45 all the way to 300 bonus damage! This is great cus he can safely proc this with his Q.
Blackout passive allows him to safely roam adding more survivability which is always good for ADCS but allows him to prey on champs and snowball.

Blade of the Ruined King (preferred vs. Tanks):
it gives 40 AD for his skills since it's a caster.
25% AS which works well with his passive.
12% life steal for survivability
The 8% of target's current health gives burst to his initial Q and allows him to wither down high health enemies.
The Active can be used to help kite and cripples low mobi champs drastically or allows you to catch up to champs who may have an escape.

5. Maw of Malmortios (Vs. AP assassins)
It gives 55 AD for his skills since it's a caster build.
10% CDR benefits him because he is a caster build.
40 MR to defend him from ap burst and synergizes well with Gage if you chose to go that route. I would get both these items in the case that the enemy team is AP heavy and has assassins.
Lifeline passive gives survivability to AP burst via shield and life steal/spell vamp. It also gives a very small buff to damage but when considering the 55 AD from the item it gives a total of 75 AD.

6. Youmuu's Ghostblade. (vs squishies)
+18 lethality gives him a boost against squishies such as other adcs and most mid laners.
It gives 55 AD for his skills since it's a caster build.
10% CDR benefits him because he is a caster build.
Passive allows him to roam and catch up to team fights and engage on squishies and run away before being engaged on.
Active give the same benefits.

I personally prefer duskblade.

7. The Bloodthirster
Pretty straight forward.
It gives 80 AD for his skills since it's a caster build.
passive of a 50-350 shield gives survivability to burst.
20% lifesteal is huge and his Q's are considered AA's

8. Manamune
Pretty straight forward.
25 AD, small buff.
250 mana for skills
passive gives mana sustain.