For most people, multiplayer First Person Shooters are fairly standard affairs: you spawn in, you run around a bit until you find the other team, and then you shoot them a bunch of times. No other role in Overwatch personifies this better than the Offensive characters, all of whom have abilities that help them move around the map faster and deal a lot of damage in a short period of time. From Tracer to Soldier: 76, this makes them great at harassing every other role, especially if you can get behind the other team.
Compared to all the other heroes in Overwatch, Soldier: 76's most memorable trait is that he is rather undeniably generic. Like your standard FPS character, his abilities consist of running at a brisk jog, firing some missiles from his automatic rifle, and laying down a healing station that can also heal his teammates. His most exotic ability? He can turn on an aimbot, which more or less lets you fire in the general direction of an enemy and kill them. With such a relatively bland set of abilities, you may be tempted to select a more interesting character but in truth, his vanilla nature works in his favor.
Consider this: despite his generic nature, he is the only Offensive character who can help healers keep the entire team alive (which in turn helps your team keep up their momentum), he has the longest effective range of all the Offensive characters (which makes him a perfect compliment for Reinhardt and Mercy), and his rifle is naturally pinpoint accurate (at least if you fire in 3 round bursts). While he's not the fastest Attacker, which makes it harder for him to harass snipers and support characters. The fact that his only real weakness is close quarters combat makes him a solid choice for every scenario, especially if you're using his Helix Rockets liberally against Torbjorns and Bastions.
If certain characters in Overwatch are caricatures of popular tropes, then Reaper is certainly one of them. Like an angsty teenager, his choice of attire and quotes makes him rather hard to miss, which means that his abilities rely on a degree of map knowledge. Contrary to how you may see him being played, Reaper is NOT a frontline character, meaning that you should never, ever be leading a charge into anything. As strong as his dual shotguns are, they are still woefully short ranged, so you must get close to enemies before you start shooting. Thankfully, his Wraith Form and Shadow Step abilities help him get across the map relatively stealthily, but it should be noted that he is incredibly vulnerable and more or less announces his presence upon arriving at a destination.
In other words, don't bother teleporting in front of or next to whatever you're trying to harass because chances are that they'll shoot you to death before you get a chance to react. Instead, teleport near them and walk the rest of the way. An annoying Widowmaker is on a roof somewhere? Teleport into a room underneath them and take the stairs. Bastion is in a corner? Go above them. Reinhardt closing in? Turn on Wraith Form and run away, then shoot him in the back. As one of the few skirmishers in the game, he relies entirely on guerrilla warfare to be effective. Most importantly, wait for your team to be alive to use them as a distraction.
Filling Overwatch's "space cowboy" role, McCree is realistically less of a cowboy and more of a guy who jumps out at you from alleys and throws sand in your eyes. As powerful as his revolver may be, it takes a significant amount of practice to be able to use it at long range. Fortunately, he is more or less a soft counter to all Tank characters in the game, capable of stunning them and unloading his gun faster than most of them can react. Simply throw his Flashbang at someone, Fan the Trigger, do a Combat Roll, and Fan the Trigger again. Normally, Fan the Trigger expends all of the bullets in his revolver, but upon doing a Combat Roll, his weapon magically reloads itself (which confers an obvious advantage).
Naturally, this makes McCree less suited for fighting multiple enemies, but he is incredibly lethal in 1 vs. 1 scenarios.
From her jetpack to her primary weapon, Pharah is a lover of all things rocket-based. Needless to say, being the only character who can consistently fly, she has an obvious advantage over every Tank and Support character. Just take to the skies or rooftops and fire away at them with impunity, and pursue as needed. With the hefty damage that her rocket launcher does, you can also take potshots at turrets and Bastions, but it is probably a good idea to engage them only if there is cover nearby, rather than floating in the air and making yourself an easy target.
Unfortunately, the fact that she can loiter in the air makes her a very high risk, high reward character, as any Hanzo, Widowmaker, turret, or attentive Bastion can shoot her out of the sky with little consequence, especially when her rockets have such a slow travel time.
The only Overwatch character to willingly bring a knife to a gunfight, Genji is the hardest Attacker to master for a reason. While he may have his shurikens, they are not hitscan weapons, meaning that it can be very hard to hit fast targets or anything that's further than 10 feet away. Fortunately, he does have a Swift Strike ability that can help him close the gap with someone and deal a fair amount of damage, though it may be disorientating to use at first. Unfortunately, he also isn't capable of surviving much damage, and his melee isn't particularly damaging.
This means that he is utterly useless against Tanks, open combat, and maps that don't have much cover or movement options. To play him effectively, you must use him more like a Spy from Team Fortress 2 and get behind the other team as much as possible, picking off their Support characters and running away before anyone notices. Remember: your ultimate ability only enhances your melee by so much, and even if you chain it with Swift Strike, you still have to kill them to reset the cooldown. And as fun as it may be to reflect Bastion's gunfire back at him to instantly kill him, you only have a three second window before your Deflect goes on cooldown.
Easily the most prolific character of Overwatch, Tracer brings an equal amount of charm and annoyance to the battlefield. Like a mosquito, she can buzz around every character and prick them to death with her guns, and if she gets hurt (or falls off the map), she can just Recall and regain the health she lost. And like a mosquito, it would be in her best interest to not get caught and smashed, especially considering her low health. Of the three skirmishers (Reaper, Tracer, and Genji), Tracer is probably the most forgiving of the three simply because she can dance around every character if played right, though some are obviously more dangerous than others.
On the bright side, her Blink ability lets her cross most of the chasms in the game, letting her bypass certain chokepoints (e.g. Objective B on Hanamura), letting you attack their Support characters from all kinds of new and exciting angles and locations that they normally wouldn't expect.