The Champion is probably one of the least-sexy archetypes in the Player's Handbook. With a scant few paragraphs, it is easily eclipsed content-wise by the Battle Master, Eldritch Knight, heck even the base fighter class itself.
But underneath the bland exterior is a veritable rock star. Let us explore the Champion.
Critical Hit Frequency: at 3rd level, you crit on a 19. At 15th level, you crit on an 18. This can be a substantial benefit -- remember that critical hits double all dice (but not modifiers) rolled.
Discussion: Well this is the main offensive benefit of being a Champion. More crits = more damage output. The Champion build discussion below will focus on how to best maximize this by getting even more crits per attack, more attacks (and thereby more crits), or more damage per crit. Getting advantage is therefore paramount -- because advantage doesn't just double your chances to hit, it doubles your chances to crit.
Athletics: at 7th level, you get a 50% proficiency bonus on any Strength, Dexterity, or Constitution check that doesn't already use your proficiency bonus, and your running long jump is (STR Mod x feet) longer.
Discussion: This seems like a lukewarm poor man's jack of all trades ability until you consider that this covers every physical skill check out there. Thirsty in the desert? Check. Walking along a narrow cliff? Covered. Taking a physical-stat character and making him even better at physical skills is a great example of maximization.
Additional Fighting Style: at 10th level, you get another fighting style. Boom.
Discussion: Everyone can benefit from another fighting style. This ability can be used either for more versatility or to enhance your core party role.
Survivor: at 18th level, while under 50% hit points but over 0 hit points, you gain (5+ CON Mod) in hit points back at the start of your turn.
Discussion: This keeps you going, makes you more self-sufficient (and the envy of every other frontline damage sponge), and with a big pool of hit points, you extend the usefulness of this ability by having a bigger range of hit points for it to activate.
Overview & Analysis
The Champion gets enhanced offense from critical hits (and certain fighting styles) and enhanced defense via certain fighting styles and the capstone ability.
Getting critical hits is therefore key, and engineering critical hit opportunities (either yourself or via allies' abilities) is paramount to take advantage of your offensive opportunities. Having the fighter multiattack base ability -- ending up with an obscene 3 attacks at 11th level and 4 attacks at 20th -- only makes the critical attacks even more frequent.
The archetype overall suggests a highly physical character, and given the base fighter class' extra ability score improvement feature (getting two more stat bumps / feats than every other class), the Champion is in a position to have very high physical stats by late game.
This in turn suggests that Champions should pair their racial selections with their intended build, so as to maximize their physical stats. Dwarves, elves, stout halflings (yes!), humans, dragonborn, and half-orcs can all make good Champions subject to wise build selections.
Two-Hander: The theme of the 2H Champion is Big Crits -- maximizing the damage done on a critical hit by using a big weapon like a greataxe, maul, or greatsword. Because critical hits double dice, crits mean an extra 1d12 - 2d6 damage when using a two-hander. The Great Weapon Fighting style is a must-have, increasing your base and critical damage significantly. This character occupies an off-tank or melee damage role.
Half-Orcs, with their Savage Attacks ability, make excellent 2H Champions -- particularly with a Greataxe, creating a 3d12 + mods critical hit. The extra critical chance afforded by Champions is arguably stronger than even half-orc barbarian's rage, and can create excellent burst damage opportunities. Dwarves, Humans, and Dragonborn also make good two-handed champions, focusing on damage output with big weapons.
This is a great style choice for players that really want to pump up their stats. Only a few feats are truly "necessary" for this style -- so boost that physical potential!
Fighting Style Options -- GWF, Defense, Archery (for variety)
Multiclass options: Rogue (more crit damage via sneak attack!), Barbarian -- reckless attack for instant Advantage and brutal critical can yield even more critical damage.
Feats: Great Weapon Master, Charger, Mobile, Resilient, Tough. Great Weapon master will make even non-critical hits feel special. Oh yes. Pound them with your greataxe until they beg you not to stop.
One-Hander & Shield: This character (offensively, anyway) is all about getting more crits -- via self-generated advantage created with the Shield Master feat and the shove action it affords. Grab a shield, use your Athletics to knock down some enemies, and enjoy the extra crits that advantage garners you.
This style also favors a tanking role, based on a high AC with heavy armor and a shield. Being a more feat-heavy role, and with less benefit from DEX than other types, you may want to focus strictly on STR and CON and spend the rest of your Ability Score Increases on feats.
Humans are obvious fits for this Champion type, with access to a feat at first level. Dwarves and Dragonborn are also naturals here with their STR and CON bonuses and other resilience-based racial abilities. Thrown weapons (with their STR focus) are a natural for One-Hander Champions.
Fighting Style Options -- Protection, Dueling, Defense
Multiclass options: Not recommended. You really benefit from the Survivor capstone ability. If you must multiclass, consider the Devotion Paladin (more defensive / healing / smite abilities)
Feats: Shield Master, Sentinel, Tough, Heavy Armor Master, Resilience, Mounted Combat
Two-Weapon: The TWF Champion is built around more crits via more attacks. Because critical hits double damage dice (not mods), you want to use the biggest off-hand weapon you can.
This style is superficially similar to the 2H Champion -- offensive warriors focused on damage output. But the TWF Champion is about getting more attacks in -- having a higher damage output via more hits but less burst damage. Also known as death by a thousand cuts. Because (at present) the style's primary benefit is the extra attack, this Champion is somewhat front-loaded to lower level play and is eventually outclassed in damage output by the 2H Champion, who is doing consistently bigger blows.
Because this style often (but not always) uses finesse weapons, high DEX characters (like Elves and Stout Halflings) in medium armor can make good TWF Champions, but with the Dual Wielder feat a high DEX is not required.
Fighting Style Options: Two Weapon Fighting, Defense, Archery (for variety)
Feats: Dual Wielder, Medium Armor Mastery, Mobile, Mage Slayer, Defensive Duelist
Multiclass Options: Rogue, Ranger.
Mounted: Often overlooked, the Mounted Champion "cavalier" (much like the One-Hander & Shield Champion) with the Mounted Combatant feat can generate his own critical hits simply by being mounted and attacking an unmounted opponent. While this is dependent on (a) having a mount and (b) being in a combat where the mount can be brought to bear, it is potentially very powerful, particularly when you consider that Lances do 1d12 damage. With advantage, a 15th level Champion gets a crit 30% of the time, and with three attacks per round, that's almost a guaranteed critical hit every round. And a half-orc Champion cavalier does 3d12 + mods on a crit ... with a shield in the other hand ...
Most Mounted Champions are built like One Hander Champions, but with Animal Handling skills and a feat focus on Mounted Combat early. The biggest challenge is finding (and keeping alive!) a worthy mount with this character, and being useful when the combats move indoors or underground.
Mounted champions benefit from using lances, which are heavy weapons and exclude small characters. Humans, dwarves, elves, half-orcs, and dragonborn make good cavaliers.
Fighting Style Options: as per 2H Champions or One-Hander Champions.
Multiclass options: Paladin is an obvious fit for Find Steed, though staying with Fighter allows you to have the full benefits of both Mounted and 2H or 1H fighting.
Other Styles: The Champion Archer is probably outshone on a regular basis by a Hunter Ranger due to Colossus Slayer and the Hunter's Mark spell.
Don't write off the Champion. Nobody benefits more from critical hits and (with the possible exception of barbarians) raw physicality as the Champion.