Every day, we see posts asking about build recommendations and or rants about their champions sucking, so I want this post to be a work in progress, and the progress can be recorded in a sticky post by our mods or more knowledgeable members, so people who are interested in whom to build, how to build, and why their awesome champion "sucks" can come to this post. 

1. First of all, not a rule, but a generally good advice, IMO. When I look at a champ, I go to HellHades or Ayumilove site to see their multipliers and kit. Multipliers give you a good idea when you wonder how much damage the skill would do. 

2. Identify the role of the champion. Whether you intend to use your champion is a speed champ, a nuker or a debuffer will determine the stats you need to focus on. 

Arena Turn meter booster - generally goes first in the Arena, so you need them as fast as possible. They often come with Increase ATK boost in the same skill. Arbiter, Seeker, Gorgorab are good examples, for Increase defense option, Doomscreech and Tarshon are great options. 

General Speed/Turn meter booster. In general content, their job is to provide speed and turn meter boost while also staying alive. It helps when they have utility other than turn meter boost, like Arbiter's and Gorgorab's revive, Tuhanarak's and Apothecary's heal etc. but some provide value by affecting enemy's turn meter (High Khatun, Diabolist).

Debuffers. There are many debuffs of different types, but more commonly, the term "debuffer" is used to indicate a champion who places Decrease Defense/Weaken debuffs. For maximum efficiency, these debuffs should go before your damage dealer gets a turn, so speed is a factor. In order to place those debuffs, your champion also requires Accuracy higher than the opponent's resistance. In Dungeons, that's usually determined by multiplying dungeon level by 10-12, in Arena, it's determined by your opponent's resistance, but requirements are generally much higher (350+). Notable AOE defense debuffers are Stag Knight, Ragash, Madame Serris, Warmaiden.  For single target, Rhazin Scarhide and Belanor are great examples. 

Buffers add buffs to your team, no accuracy is needed, but you generally want them to go early, so speed is important. In Arena, a turn meter booster will often provide an attack buff early on. With shield buffs, you want stats that affect shield (Defense for Valkyrie and Helicath, HP for Krisk, Blind Seer etc.), other buffs are generally not affected by stats. 

NOTE: With buffs and debuffs for content other than Arena, it's worth paying attention to the ratio between buff/debuff duration and cooldown. You want to keep buffs on your champions and debuffs on your enemies as much as possible, which is also where buff/debuffs extenders/reducers come in (Godseeker Aniri, Demytha, Lanakis the Chosen etc.)

Healers. You want them to go often, you want them to stay alive, and you want to pay attention what their heals scale off (target max hp, their own max hp, damage taken, damage dealt etc.) as well as natura of the heals (instant heals, continuous heals, damage heals (Leech) etc.). Apothecary, Mother Superior, Runekeeper Dazdurk, Vogoth, Lugan, Wythir are a few examples.

Cleansers remove debuffs, very helpful in every content where you face debuffers. Runekeeper Dazdurk, Spirithost, reliquary Tender are helpful cleansers. Build them fast and tanky.

Crowd Control champions are generally champions who place debuffs that dictate/cripple enemies' possible moves. This means that CC champions need to be built with high ACC, Speed and in case of provokers - tanky. The best means of crowd control is Stun (Ragash, Big 'Un, Miscreated Monster, as well as Stun Set on AOE champs) and Provoke when provokers take advantage of taking hits, like Maulie Tankard, Giscard the Sigiled, and pair well with Counterattack buffs. Block Active Skills and putting active skills on cooldown (Lugan the Steadfast, Bellower, Warlord) work similar to Provoke but with less direction. Freeze and Sleep (Tormin, Kymar, Seer) work similar to stun in that they make the enemy skip their turn altogether; however, Freeze reduces damage dealt to them by 25% while Sleep is removed by an inflicted hit. Another form of Crowd Control is turn meter manipulation (Psylar, High Khatun, Rhazin Scarhide) and Decrease Speed debuff (Tuhanarak, Stag Knight, Psylar).  

Damage Dealers: this is an entire section on its own, and where many make mistakes. Generally, there are 2 types of damage dealers - raw damage and debuff damage. 

Raw damage dealers can be scaling off your champion's ATK, DEF and HP or Enemy Max HP, which is much more valuable in higher level dungeons and bosses due to the enemies' high HP pool which scales the damage. In the case of the former, they should be built with high stats affecting the damage, while the latter do not generally benefit from them. ANY RAW DAMAGE DEALER SHOULD BE BUILT WITH 100% CRIT.RATE AND AS MUCH CRIT.DAMAGE AS YOU CAN PUT ON THEM. Seriously, I can't stress this enough. I have clanmates talk about their nukers not hitting hard even though they have 7K attack. Every time that happens, they have ATK% gloves on and no crit.rate or crit.damage. 

Debuff damage includes Poisons, HP Burn, Bombs etc. Like any debuffers, they need to be bulilt with Accuracy to land them. In case of Bombs, they scale off ATK while poisons and HP Burn scale off enemy max HP. 

Protectors - well, they protect using buffs like Ally Protect, Veil, Strengthen, Increase Defense etc. Duchess Lilitu, Ragash, Miscreated Monster, Krisk are excellent examples. Build them with a lot of resist, a lot of tankiness, and Speed.

Revivers - build similar to protectors because they need to outlast the rest of your team in order to bring them back. Cardinal, Ursala, and Duchess Lilitu are great examples.

3. Once you identified the role your champion will play, you'll need to gear them out. Naturally, be aware of the possible main stats on each piece. Speed can be found only on Boots, Accuracy and Resistance can be found on Chest and Banner, Crit.Damage can be found on Gloves and Amulet, Crit.Rate is found only on Gloves. 

Look out for overlap between stats and sets, but generally, stats come first. As such, if you find an Accuracy set item with Accuracy stat/substat or a Speed set item with Speed stat/substat, take that into consideration. 

Notable gear pieces are:

Speed boots - valuable almost everywhere on almost every champ.

Crit. Rate gloves - good way to boost your crit.rate to 100%

Crit.damage glove with crit.rate substat - must have for raw damage dealers when combined with other pieces that provide enough crit.rate.

Accuracy Banner - for obvious reasons

Banner with Speed substat.

Crit.Damage amulet for damage dealers. 

Resistance banner for protectors, cleansers, revivers etc.  

Savage and Lethal sets for damage dealers

Perception set for debuffers

Lifesteal and bloodthirst sets for when your squishy damage dealers need to stay alive. 

Stun set to turn every AOE champ into a potential crowd control (works best with all-AOE champs like Psylar and Bellower as well as AOE A1 champs like Soulbond Bowyer and Deliana)

Like I said, add what you think is needed and someone can post and update the collective edited information as we go.