Ezreal is, in my opinion, one of the hardest champions to play because he relies entirely on skill shots to get his damage off especially in the more defensive builds. However, if played correctly, he can be very rewarding and effective. His laning phase is not too good but mid to late game, his damage is insane and the slow from the Iceborn Gauntlet makes it easier to hit successive abilities on targets and output tonnes of damage.

For the runes, I suggest taking quints and marks of AD and then magic resist blues and armor yellows because this allows you to be harder to take down while also giving you a lot of early game kill potential. The keystone i use is Thunderlord's Decree because this makes his burst much better and works well with his e, w, q combo. The extra damage is good for poke in lane but also very effective if you are building more hybrid. The masteries also give you extra ap and ad per level which also increases damage and allows you to build more ad or more ap depending on what the enemy team is building.

The Manamune is the core item for this build and you should rush this first to stack it up as quickly as possible. The second core item is the Iceborn Gauntlet and this should also be rushed as it provides mana which effectivly gives you more ad because of the Manamune but also gives CDR. After that it is up to you. For example, if you are against a tanky team then building Blade of the Ruined King is good as well as Lord Dominik's Regards for the percentage health and armor penetration.
Personally, I like building the Hextech Gunblade as this gives you aps ap as well as ad. It also gives good sustain from the 15% of damage dealt returned as health. Combined with the Spirit Visage it can give some really good healing in a fight which allows you to stay alive longer and is especially effective verus ap/tanks like Ekko.

Mystic Shot is your main damage ability so you should max this first. For the second ability, I prefer to max Essence Flux as this gives increased attack speed which is really useful in a 1v1 or in teamfights. it is also good harass as it can pass though minions. However, this can be swapped out for Arcane Shift if you favor this ability. For summoner spells, I normally take Heal and Flash for bot lane as these are the best ones to have in a 2v2 however Heal may be swapped for Ignite if you are a more aggressive player.

For the laning phase, you can use Mystic Shot to easily out poke your opponent and farm. Arcane Shift allows a quick escape but you will be vulnerable after using it. Try to use your passive to your advantage by getting to 3 or 4 stacks before engaging the enemy. Good supports to have are Bard, Leona or Morgana as they all have stuns or snares which makes it easier to hit your skillshots.