Check notes for explanation of items. Might fix format someday. Skipping the "basics".
IF you are looking to onetrick this champ here are some of the big downsides you should simply accept.
If the enemy can dodge you do zero damage.
You wlll be relying on your teammates and abuse vision.
You will not be able to use splitpush as a wincon without outside help.
"You should not try to be the hero."~ Xenonym, 2014 Rank 3 Xerath on mid/supp

The last point is really what sets you apart from preplat and afterwards. Before platinum people won't be able to dodge as much. This means you can simply rush through those brackets with a mejais and massacring everyone as your damage is very inflated. However afterwards people will start dodging your abilities. This means you have to adapt.

E buff allows you to play much more aggressive. You can now melee E and get out.

#
Aim at the feet.
Aim between their current position and their walking direction.
Abuse Corners, Running away and the spot between tower and wall.


P - Mana Surge
This should be on cooldown. Don't suicide for it. Place a ward on your own Birds for mana on rainy days. For this a tethering guide may be of use: Play Video
Q- Arcane Pulse
Main waveclear and Poke skill.
Range indicator for yourself is accurate. However for enemies it can be deceptively long. Abuse Brushes and Fog of war for free poke during lane and during midgame when waves crash.
You can move while charging, but get slowed, which makes it a bad choice for chasing.
W - Eye of Destruction
Setup Skill, Waveclear helper.
This skill has extra range due to being AoE. Try to wait on other people's CC if possible.
Refer to Early game instructions and combo.

E - Shocking Orb
Setup, Self protection
Offensive use when in Killrange for your combo. Defensive use to prevent enemy engage. Alternatively use it after the MELEE enemy finished their combo and is disengaging. This allows you to put quite a few more autoattacks into them.
Due to the buff you can now use this very effectively to get away, making it much more important than ever before to gauge whether it is worth to use it offensively.

R - Rite of the Arcane
Killcombo, Ranged Gank, Teamfight Winner

Important Map Positions
Red and Orange for Ult
Purple Points for Q
Purple Lines for E that people forget

EQ-WR(RR) - Best Combo, 2 R may miss vs people with a dash. Compensate killthreshold by hitting 2 more Qs.

WQ - Poke during lane. You waste mana if you don't hit Q.

WEQ - Setup E with W to mitigate enemy ability to dodge

EQW - Fastest Combo (Tap Q), Within E range

E-AA-QW-AA - Due to the buff you can now add another AA for free at the end.

WRRR - Yes this works.

E-RRR - Due to the buff you can now shoot 3 ult shots after and E.

WQ-ERRR - This does less damage than the first combo, but the W slow may make it easier to hit your E.

In the following I will be skipping 34 MR (ADC=, because they are basically always oneshot from a combo. Shadowflame will be set to 15 as the average MRpen.

Against 40 MR EQWRRR Damage Values:
29 AP: 740
49 AP: 792
64 AP: 831
Against 34 MR Add 50~80 Damage.
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Lvl 9 - EQWRRR DMG, Ignoring First Strike or Comet, Mythic + Boots
Ignoring Liandry Dot effect.

vs 40 MR
Ludens + Sorc Boots, 97 AP: 1359
Liandry + Sorc Boots, 97 AP; 1166

vs 50 MR
Ludens + Sorc Boots, 97 AP: 1261
Liandry + Sorc Boots, 97 AP: 1077

vs 70 MR
Ludens + Sorc Boots, 97 AP: 1106
Liandry + Sorc Boots, 97 AP: 936

Lvl 11- Boots + Mythic + 2nd

vs 40 MR
Ludens + Shadowflame, 197 AP: 2140
Liandry + Shadowflame, 197 AP; 1865

vs 50 MR
Ludens + Shadowflame, 197 AP: 2026
Liandry + Shadowflame, 197 AP: 1705

vs 70 MR
Ludens + Shadowflame,, 197 AP: 1724
Ludens + Void Staff,, 162 AP: 1845
Liandry + Void Staff, 162 AP: 1550

vs 90 MR
Ludens + Void Staff, 162 AP: 1688
Liandry + Void Staff, 197 AP: 1420

3 item Killcombo DMG isn't usually relevant due to threat changes and damage being high enough for the relevant people.

----
Other relevant numbers:
----
Stats: lvl 11 371 AP, 34 mr pen Items: Luden+Sorc boot+Rabba+Rylais
Use R three times vs 34 MR: 1413 Damage
Use R four times vs 34 MR: 1830 Damage

Use R three times vs 50 MR: 1235 Damage
Use R four times vs 50 MR: 1595 Damage
---
Stats: lvl 14 409 AP 34 mr pen Items: Luden+Sorc boot+Rabba+Horizon+
EWQ against 34 MR: 1775 Damage
EWQ against 50 MR: 1530 Damage

Early Game
The classics: Aim on Feet. W-Q when the AA minions or use skills. Proc Passive while they are slowed. AA them while they are stunned with E for extra dmg.

LvL 1 Q against pushers. Q the wave and AA minions hard.
Lvl 1 W against non-pushers. Wait on First Strike or Arcane commet for usage.
Lvl 2: Choose whether you want to setup for gank or contest grab. Defensive: Keep mana high. Offensive: Start poking with WQ+AA
Lvl 3: Gank + kill with E or scale.

Lvl 6: Use Killthreshold to setup kills. For simplicity you can assume that 2~3 qs are enough.
Continue scaling and look for ganks on Sidelanes.
It takes roughly 8 seconds from top to mid and from mid to bot. Know this to know when the jungler is arriving.

Early Game decision making:
1. Check Threats
> Far threats = Sidelane, krugs or frogs
> Near threats = Between far and direct
> Direct Threats = On lane, 1 Brush away
2. Check Opportunities
> Ult + Ally Kill Threshhold - Possible under Direct Threat on opposite side
> Wave state allows Gank - Shove and Gank (Top for snowball in melee matchups)
> Reset for resources - Useful when you are low otherwise stay
> Plating gold - Risky if you don't know enemy jg position, Follow your laner or reset
> Scale - Csing is sometimes simply the best choice

Waveclear:

Lvl 9: Max lvl Q.
Multiple methods may work:
Hit sorc with W and Q. Lasthit fron with AA. - Safer than second
Hit melee + sorc with W and Q. AA front 1~2 times. - Front HP differences can be problematic, beware of your position to enemy laner
Hit Q on wave, W-center frontline. - The most comfy and safest. Wait a bit for front to lose hp.

Now some math for third method:

Get 117 AP to oneshot Sorc minions with Q.

For perfect Xp we reach lvl 9 at minute 10, which means lvl 6 HP upgrade for minions => 328 HP -> You need 116 AP for a 329 Damage Q.

Luden: +9 Rune, +15 Doran , +40 Lost chapter, +40 Blasting Wand, +15 Dark Seal = 117
Crownguard: +18 Rune, +9 Absolute, +15 Doran , +40 Lost chapter, +20 Amp tome, +15 Dark Seal = 117
Liandry: Just Blast.

Midgame

Vision -> E -> Kill -> objective
Look at wave state -> Figure out which wave will crash into you -> Take it.

Slightly less obvious things perhaps:
30 seconds before Epic Monster you should press B to return to base. You will return on time.
Control wards are only as good as the people who protect them. More defensive usage of them could be making more out of the invested gold. Place them in your ult spots for example.



Lategame
Choose your target for Q poke is posible. You can carry with Xerath in 2 ways. Kill the carry or remove the frontline. Sometimes the lulu is more important than the 20/0 Jinx.
2 ~ 3 Qs should kill most squishies.
EWQ should kill most squishies.
Ult should kill most squishies. Wait for others to hit a CC or engage then blast.
Poke with Q is slightly more problematic. You should try to hit from outside of their screen.

Xerath can carry in lategame by zoning people out of the fog of war with pure damage.

• Sometimes you have no killpressure (yasuo). Accept it, don't waste mana on poke and focus on wave.
• Previous point can be rendered irrelevant depending on your jungler and support.
• Plating + Wave tends to be more valuable than following a gank. Careful of jungler tho.
• Danger Ping on top of sidelanes and ping missing midlane. Harder to not see a red attention mark on the screen.
• Place a stealthward near the enemy turret to see in what direction he is roaming.
• Being unlucky with teams is just a fact of life at this point. Simply try to get better at turtling or playing risky in those games. Still good training.
• Sometimes you simply lose in the draft. In those cases, dodging may be preferable for your mental or you can find a way to win and improve instead.

8 years deathiversary soon. Twin Shadows made this character so much better at carrying so hopefully we get it back next year. :/