Welcome to my Kayle guide, where you will learn to build her as a hybrid DPS. Yes, Kayle can be built tanky or as a support. However, I have found that Kayle is best built as a hybrid DPS, so she is capable of dealing mass amounts of damage while being able to somewhat support teammates in lane and team fights. Let's take a look at her pros and cons:

Pros:

Can heal HP
Can speed allies/slow enemies
Can be built into many roles
Can grant invincibility (though very short)
Can deal splash damage

Cons:

Somewhat squishy early game
Has MP problems early on (depending on who you're laning against)
Needs to be farmed

My runes are as follows:
9x Greater Mark of Insight
9x Greater Seal of Replenishment Greater Seal of Replenishment
9x Greater Glyph of Focus
3x Greater Quintessence of Potency

For my marks, I chose Magic Penetration so you can still deal decent amounts of damage, even against a tanky Galio (maybe..).
In my seals, I have flat Mana Regen per 5 because of Kayle's early-game mana issues. The reason I have flat over per level is her mana problems diminish later on in the game, especially since I recommend taking golem if you don't have a different AP carry.
For glyphs, I chose flat CDR because her abilities will need cooldown reduction in the laning phase to have more of an effect on allies and foes.
Finally, quintessences are flat AP, because they help Kayle's early game heal against harassment and Reckoning for harassing.

I take the standard 9/0/21 Caster build for Kayle. Grab the CDR and MPen in the offense tree to increase early-game power. In the utility branch, grab the death timer reduction, EXP boost, and mana regen boost. After that, build towards your Presence of the Master (final Utility mastery) while grabbing your two summoner spells' masteries.

I cannot emphasize your early-game mana issues enough. Grab Meki Pendant (opposed from Mana Manipulator, but if you really need to.. go ahead) and 2 HP pots. This will keep you in lane long enough to farm creeps just enough to afford your brown boots (Sorc Shoes) and upgrade your Meki Pendant up to a Fiendish Codex.
Soon after that, you want to upgrade your Tier 1 boots into Sorc. Shoes(or Merc Treads if they're CC-heavy) if you haven't already, and you also want to get a Guinsoo's Rageblade (or at least start building it).
If you prefer rushing Nashor's Teeth over Guinsoo's, go ahead, but because Guinsoo's is a hybrid item, you'll be boosting your skills' effects as well as your regular attacks (especially with Righteous Fury). But in most cases, you'll be grabbing Nashor's after Guinsoos.
If you can't wreck them fast enough or are killed before you can do enough damage/support, grab a Hextech Gunblade (or at least Hextech Revolver) before going into other CC-stopping, Tryndamere-killing, tank-shredding items.
If your game hasn't ended from your wrecking ball of Hybrid DPS and support already, start grabbing your next two items. Your previous four items are considerably more expensive and more effective, so you should be destroying after grabbing your fourth item. Here are the guidelines for your last two items:
General (can't decide) - Zhonya's Hourglass AND THEN Rabadon's Deathcap
CC-Heavy - Banshee's Veil AND/OR Quicksilver Sash
Tanky/Beefy - Madred's Bloodrazor AND/OR Void Staff AND/OR Last Whisper* AND/OR Sword of the Divine*
Dying Too Fast - Rylai's Crystal Scepter AND/OR Zhonya's Hourglass AND/OR Rod of Ages*
Tryndameres (escaping) - Rylai's Crystal Scepter AND/OR Lich Bane*
( * = decent but not recommended )

Holy Fervor- 30% of Kayle's attack damage is added to her ability power and 15% of Kayle's ability power is added to her attack damage.
This spell is what makes Kayle superior as a hybrid DPS. Grabbing AP items will benefit your AD, and grabbing AD items improve your AP. That's why buying items such as Guinsoo's Rageblade and Hextech Gunblade make Kayle stronger with spells and regular attacks.
ReckoningBlasts an enemy unit with angelic force, dealing damage and slowing movement speed. While the target is slowed, Kayle inflicts increased damage to them.
Kayle is great at early game harassment and chasing because of this one skill. If you build a flat AP rune page, you are guaranteed to hold your enemies at their towers for the laning phase. Even without a burst of AP, this spell allows you and your lane partner to follow up with a series of CC, spells, and regular attacks.
Divine BlessingBlesses a target friendly champion, instantly healing them and granting increased movement speed.
This spell needs to be used as often as possible to keep your team strong. Because you are still part-support, you will also need AP for this skill, not just Reckoning or Righteous Fury.
Righteous Fury Ignites Kayle's sword with a holy flame, granting Kayle a ranged splash attack and bonus magic damage.
This is your first priority after your ultimate. It allows you to farm minions, harass opposing laners, and kite enemies (with Reckoning). This completes Kayle's hybrid DPS.
InterventionShields Kayle or an ally for a short time, causing them to be immune to damage.
Hallelujah! Now this is what I'm talking about! It gives an Endless Rage for 2/2.5/3 seconds. This will need to be saved for your opponent's focus (most likely the Carry), but you can give this to anyone else if they really need it. This is best used if followed up by Divine Blessing.

Your summoner spells:
Flash Teleports your Champion to target nearby location under your mouse cursor.
I recommend this spell because it helps you get out of ganks, team fight misplacement, catch up to enemies, and catch up to allies who need healing/invinc.
Clairvoyance Reveals an area of the map for 6 seconds.
I really recommend grabbing this because you are still a support, and MAP AWARENESS WINS GAMES.
GhostYour Champion ignores unit collision and moves 27% faster for 10 seconds.
This spell used to take the slot of Clairvoyance in my build, but I have found that Flash is sufficient enough to get out of ganks, chase, etc.
Ignite Ignite is a damage over time spell that targets a single champion dealing 50 true damage plus 20 true damage per level over 5 seconds. This damage is not lowered by armor or magic resistance. Also reduces the target's healing and regeneration by 50%.
I wouldn't really recommend this spell because Kayle has a slow, a speed-up, and a long-range attack. You could use this if you encounter many stray, low-HP moments.
CleanseRemoves all control debuffs, silences and blinds from your champion and reduces the duration of subsequent stuns, slows, taunts, fears, snares, sleeps, and immobilizes by 65% for the next 3 seconds.
This spell is useful if you are being focused down with all of the team's CC. But, I doubt that they will choose CC-ing you over other DPS/carries on your team. If you insist, go ahead.
TeleportAfter 4 seconds, teleports your champion to target friendly minion or turret. 3 minute cooldown if cancelled.
This spell is great to help you go around the map. I don't really recommend for Kayle to hold the teleport on your team because she is better partnered up than alone farming minion mobs.
ALL OTHER SUMMONER SPELLS ARE NOT RECOMMENDED BUT ARE STILL AVAILABLE IF YOU CHOOSE.

So now you know how to build Kayle as a hybrid DPS. Sure, you can build her as a big, oily tank or a tiny, flimsy support.. but to become a wrecking ball, you should build her as a hybrid DPS. Kayle isn't very popular in League of Legends, mostly because people are afraid of owning people too hard. Just imagine...

You can have:
JAX (hybrid dps)
TARIC (heal)
ZILEAN (speed/slow)
TRYNDAMERE (invincibility)
all into one champion. Or you can just pair yourself up with one, only ONE, of these four. But after all, four is greater than one!