Pros:

• Has some of the quickest and most reliable burst in the game.
• Is an amazing splitpusher and can take towers down incredibly fast with her passive + itemization.
• Many items are viable on Diana.
• Is somewhat durable.
• Has great mana sustain in lane.

Cons:

• Has a relatively weak early game against many matchups.
• Is susceptible to ganks.
• Has no health sustain
• Has no escape.
• Is somewhat weak in team fights.

I usually go 12-18-0 on Diana, though you can play around with 0-18-12 masteries on Diana (some are pretty good on her, but they're more defensive).

Fury VS Sorcery: I personally prefer Sorcery for the bonus burst damage. Fury is also viable due to Diana's passive, but a bit weaker in my opinion.

Fresh Blood VS Feast VS Expose Weakness: Fresh Blood is the best choice. It further adds to Diana's burst. Feast is also viable against difficult matchups. Expose Weakness is the worst choice.

Vampirism VS Natural Talent: Natural Talent seems to be the best pick. The extra AD is nice for last hitting and it doesn't go to waste since Diana is a very AA-reliant champion.

Bounty Hunter VS Double-Edged Sword VS Battle Trance: I'm currently preferring Battle Trance. It helps Diana substantially in the early game.

Wanderer VS Savagery: I prefer Savagery as it makes last hitting easier. Diana will be fast enough by mid game.

Runic Affinity VS Secret Stash VS Assassin: I prefer Secret Stash for the extra mana sustain. Assassin is equally viable if you're better at managing your mana. Runic affinity is really only viable on jungle Diana.

Merciless VS Meditation: Meditation is very strong, but Merciless allows Diana to more easily execute her opponents. Either is viable.

Greenfather's Gift VS Bandit VS Dangerous Game: Dangerous Game is the best option, especially since Diana is a very strong diver. Greenfather's Gift is useless on Diana, and Bandit is not worth the small amount of bonus gold.

Precision VS Intelligence: I prefer Precision. Magic penetration allows Diana to kill squishies easier, and a full build Diana will usually have between 30-40% CDR. An extra 5% isn't really necessary.

Stormraider's Surge VS Thunderlord's Decree VS Windspeaker's Blessing: Thunderlord's is by far the best option. Diana can easily proc it and it does a ton of damage early. Stormraider's is an okay option against very mobile champions but I don't recommend it. Windspeaker's is bad on Diana.

Level 1: Put a point into W. Diana's level 1 is very weak and the poke from Q is pretty useless this early on. W helps Diana CS without taking too much return damage.

Level 2: Put a point into Q. Diana use Q to harass weak laners or to secure last hits on minions against difficult laners.

Level 3: At level 3, the skill sequence changes depending on your laner. Against easy match ranged matchups put a point into Q for added poke. Against very hard match ups, put a point into W for the extra shielding. Against average melee match ups or matchups with high jungler pressure, put a point into E.

Level 6: By level 6 you should have either 3 points in Q, 1 point in W, 1 point in E, and 1 point in R, OR, 2 points in Q, 2 points in W, 1 point in E, and 1 point in R.

For the rest of the game, max Q first, then W, then E, and add a point into R whenever possible. You can put additional points into W earlier if you feel like you'll need the extra shield.

Moonsilver Blade (Passive): Every third basic attack deals a bonus 20-250 + (80% AP) magic damage in a small semi-circle around Diana and restores (15% AP) mana. Casting an ability grants Diana 20% bonus attack speed on her next 3 basic attacks (50/60/70/80/90% bonus attack speed based on rank in E).

Diana can do substantial damage with her passive, so it's important that Diana uses her auto attacks as frequently as possible when safe. This allows her to often out trade champions like Yasuo. Diana's passive also works on towers. With this passive, Diana can take down towers very fast, and even faster with Lich Bane.

Crescent Strike (Q): Diana throws a bolt of lunar energy in an arc, dealing 60/95/130/165/200 + (70% AP) magic damage, and applying a debuff called "Moonlight" on all enemies hit for 3 seconds. Diana and allies gain vision of enemies debuffed with Moonlight, and Diana's ultimate is fully reset when cast on an enemy with Moonlight.

This is Diana's only form of poke. It's a strong ability. Since it is a curved skill shot, it can be difficult for some opponents to dodge. This ability is also great for scouting brush as it will give vision of all enemies hit.

Pale Cascade (W): Diana shields herself with 3 orbs. The shield absorbs up to 40/55/70/85/100 + (30% AP) damage and deals 22/34/46/58/70 + (20% AP) magic damage per orb to an enemy (66/102/138/174/210 + (60% AP) magic damage total). Diana's shield refreshes when all of the orbs are consumed.

Diana's shield can allow her to proc Thunderlord's decree very easily, and can allow Diana to with many melee trades early.

Moonfall (E): Diana pulls nearby enemies towards her, slowing them by 35/40/45/50/55% for 2 seconds.

This ability allows Diana to apply her passive more easily, but it can also be used to interrupt channels (like Katarina's ultimate or Miss Fortune's ultimate). It's great for chasing.

Lunar Rush (R): Diana dashes to a target enemy, dealing 100/160/220 + (60% AP) magic damage. If an enemy is affected with Moonlight, then Lunar Rush's cooldown is refreshed.

Lunar Rush adds a ton of quick burst for Diana and aids in her power spike.

Early Game Combos:

Level 1 Melee Trading Combo [W > AAs]
Diana can use this combo at level 1 to win early trades against melee matchups with poor level 1s (like Akali). It will also certainly proc Thunderlord's Decree, and if you're lucky, you might also get your passive off.

Level 3-4 Melee Trading Combo [E > W > AAs > Q]
This combo is also best against melee matchups. If they come close enough, pull them in with E and use W to shield some of their retaliation damage. Often times they'll realize they're losing the trade and try to run away. You can get autos off fairly easy since they'll be slowed, and then poke them on their way out with Q. Be careful to not draw too much minion aggro though!

Level 6 Burst Combo [Q > R > E > W > AAs > R]
This is Diana's burst combo. If you land a Q, follow it up with an R. The rest of the combo is pretty much the same as the one described above. If you get 3 AAs and Thunderlord's Decree off with this combo, you can do as much as 550 + (330% AP) magic damage to a target! For even more damage, you can AA minions twice then proceed to do this combo - however, this is more telegraphed and often your opponent will back away.

Level 6 Chunk Combo [R > E > W > Q > AAs]
This can be used to scare many match ups. This combo does a lot of damage and doesn't require you to Q first. It can easily catch your opponent off guard and blow their flashes. It's great to bring targets to 50% health before you all-in them when R is off CD. When you get a few items, you can actually one-shot squishies with this combo. Be careful though! Only use this combo if you're sure you're not getting baited in for a gank!

Burst Combo Against High-Range Opponents [Flash > Q > R > R > E > W > AAs]
This combo is best against squishy targets that are playing too safe for you to land Q. It deals a ton of burst, but only use this if you're confident you can execute your opponent as you don't want to unnecessarily waste your flash.

Engage Combo [Flash/Q > R > E > Zhonya's]
I don't really like wasting Flash and Zhonya's, however, if the enemy team is at a chokepoint, this combo is great for pulling enemies together for your team to follow up. It's great if there's champions like Yasuo or Malphite on your team.

Escape Combo [Q (Minion OR Nearby Jungle Camp) > R]
Diana has no built-in escape, however, if you're running away and there's a minion in front of you, you can Q > R it to gain some distance. In lane, you can Q > R raptors to make for a crafty escape. You can also use this against Gromp, Wolves, etc... (provided that they haven't been killed).

Mid/Late Game Triple R Combo:
Mid/late game when you have 40% CDR your Q cooldown will be 3.6 seconds, while your Moonlight debuff lasts for 3 seconds. You can use this to initiate in team fights if you mark a front liner as it will usually get you access to the back line. You can also use this in extended fights. The combo is Q > Wait 3 seconds > R > Q > R > R.

Tower Pull Combo: There have been MANY times where enemy junglers try to gank me at level 3 onward. If you think you might die and need to flash INTO your tower, you can sometimes Flash Away > E and pull the enemy whose diving you into your tower and secure a kill.

Starting Items:

LARGE UPDATE AS OF PATCH 7.22

I have completely changed the build path in this guide. Full burst Diana feels substantially weaker in this meta compared to AP bruiser Diana.

Starting Items: Starting with the standard Doran's Ring + 2 potions (biscuits) works well on Diana. If you feel like you can snowball of your opponent really easily (e.g. if you're against someone like a Karthus) you can go for a Dark Seal + 3 potions.

Core Item: Nashor's Tooth is (in my opinion) the best item for Diana as of patch 7.22. This item provides you with a TON of attack speed, CDR, and a great passive which synergizes well with Diana's passive. With games ending sooner, the attack speed and passive allow you to take down turrets quicker and become more of a threat. If you're against a rough AD/AP matchup you can opt to purchase Zhonya's Hourglass or Banshee's Veil first. When you back try to have enough gold to at least buy Amplifying Tome + Dagger.

Boots: Sorcerer's shoes are optimal to burst down squishies. If you're against a rough matchup you can opt to choose Ninja Tabis or Mercury's Treads.

Second Core Item: Usually there will be at least one threat on the enemy team. Try to itemize against them by building either Zhonya's Hourglass or Banshee's Veil. If your entire team is far ahead you can skip these defensive items for more offensive items.

Update: Abyssal Mask is quite strong as of 7.21. You can definitely choose this over Banshee's Veil for a more defensive option. It provides decent health, CDR, mana, MR, and gives you a passive which allows you and other AP damage sources to deal 10% bonus damage. It's a good pickup against champions like Azir.

Third Core Item: At this point in the game you'll probably need to start building some HP for survivability. At the moment, Hextech Protobelt is your best bet. It provides some additional mobility and burst with the passive.

Other Items:

Wit's End: Wit's End is an alternative to Nashor's Tooth. Nashor's Tooth provides AP for your ability burst, higher attack speed, 20% CDR, and an immediate 27 on-hit magic damage on its passive (and scales). Wit's End is 500g cheaper, provides a 40 magic resistance (up to 65 magic resistance), gives good attack speed, and an immediate 40 on-hit magic damage (does not scale), and magic resistance shred. Wit's End would be a good buy if most of the damage threat on the enemy team is magic damage and the enemy ADC is a low damage ADC and/or does a lot of magic damage (e.g. Ashe, Varus, Kog'Maw).

Void Staff: This item is essential for 1v1 bruisers in the mid/late game. It's stronger than Rabadon's at the moment.

Rabadon's Deathcap: Diana has some very high AP scalings so this is still a good purchase on her but it costs SO MUCH and provides no defensive stats, so it's somewhat of a riskier buy.

Rylai's Crystal Scepter: This item is actually pretty good on Diana. If you land a Q on your target, the slow allows you to get in range to R. It also provides a decent durability and great sticking power when your E is on CD.

Hextech Gunblade: I'm on the fence about this item. Diana will only receive the full 15% healing from her R. Her P, Q, and W will only give her the 5% healing. The strong part of this item for Diana is the extremely strong passive which gives her even more burst.

Rod of Ages: This item is core as a first/second item for many Diana mains. As of patch 7.21, this item feels quite weak on her. It delays her powerspike (Diana is strongest at 2-3 items imo) for more early game durability. This might not be a bad item if you're ahead but I'm currently opting away from it.

Liandry's Torment: This item synergizes well with Diana's E and gives her more durability with the HP and burst with the magic pen. It also lets her duel bruisers more effectively. It's not a bad pickup but somewhat situational.

Morellonomicon: This item is strong on many mages but is fairly weak on Diana. Diana has sufficient mana sustain with her passive and the stats from Nashor's Tooth are more effective for her kit.

Lich Bane: This item is great for assassinating squishies but falls off HARD as of patch 7.21. Assassin-builds are much weaker with all of the enchanter shields, heals, etc... so a more durable sustained DPS build is more effective at the moment.

Luden's Echo: Only buy this if you're far ahead as the combat stats fall off hard otherwise.

Champion: Amumu
Threat: 3/5 (Average)
Description: Amumu's early game is fairly weak, so his early ganks will only be successful if you get ganks at medium to low health.

Champion: Graves
Threat: 1/5 (Very Low)
Description: Graves' ganks are pretty bad unless your lane opponent has good lock down. Midlane doesn't have a lot of walls though, so the damage from the second part of his Q is easy to avoid.

Champion: Ivern
Threat: 3/5 (Average)
Description: Ivern loves to gank, but he's no more of a threat than Amumu in the early game. He'll be more of a pain for your jungler.

Champion: Elise
Threat: 4/5 (High)
Description: Elise's level 3 burst is high but she can't do anything to you unless you eat a cocoon to the face.

Champion: Hecarim
Threat: 3/5 (Average)
Description: Hecarim used to be a lot stronger, but since his nerfs, he's less of a threat as his damage and displacement is just average.

Champion: Kha'zix
Threat: 3/5 (Average)
Description: Kha'zix has very high early game damage but he has difficult sticking onto targets.

Champion: Kindred
Threat: 3/5 (Average)
Description: Kindred's early ganks are scary, but she'll likely mark you in advanced before ganking you. She's very squishy, so you can easily kill her later if she's caught out without her ultimate.

Champion: Lee Sin
Threat: 5/5 (Very High)
Description: Lee Sin is a very strong early jungler, and Diana is an easy target from levels 1-5. There's a good chance he'll try to gank you at least a couple of times, so be sure to ward important jungle chokepoints.

Champion: Master Yi
Threat: 1/5 (Very Low)
Description: Master Yi is a farm jungler, so the chances he'll gank you are low, and the changes he'll get a successful gank off is low too.

Champion: Nocturne
Threat: 2/5 (Low)
Description: Nocturne can't really gank mid before level 6 and will likely avoid mid unless overextended.

Champion: Nunu
Threat: 1/5 (Very Low)
Description: His ganks aren't that good. His only CC is a slow, and he doesn't do much damage in the early game.

Champion: Rengar
Threat: 2/5 (Low)
Description: Rengar has very little mobility in the mid lane as he needs bushes to be most effective. He's more of a threat at level 6, but his ganks can be highly telegraphed and avoided.

Champion: Volibear
Threat: 2/5 (Low)
Description: All Volibear can do is run at you. If you're not overextended, he likely won't kill you.

Champion: Warwick
Threat: 2/5 (Low)
Description: The new Warwick is much better at ganking longer lanes and probably won't be a big threat before level 6. At level 6, Warwick can easily miss his ultimate.

Champion: Xin Zhao
Threat: 4/5 (High)
Description: Xin Zhao's damage is high early game, and he can be a real threat if he ganks you with red buff with his slow and knock up.

Champion: Zac
Threat: 5/5 (Very High)
Description: Zac's long range engage makes it difficult to ward against him and his CC can make it hard to escape ganks. Very punishing against champions with no escapes or MS steroids