(Please note that the recommendations below are largely made for in-game play. A Clawitzer in a competitive environment will likely make use of different moves to accommodate for smarter, more adaptive opponents. If you have a different idea for a Clawitzer move set, feel free to note it in the comments.)
Clawitzer is about as predictable and straightforward as the gun after which it is named. This pokémon is a killing machine. It doesn't play tricks or mess about with fancy gimmicks, it just launches hard-hitting moves at its foes - and it does so with lethal precision. Most enemies will know what you're going to do when you deploy a Clawitzer, but that doesn't make it any less dangerous.
Straightforward leveling, here. Get your freshly-caught Clauncher up to level 37 and it will evolve into a Clawitzer.
Fire, Ice, Steel, Water.
Close to a glass cannon but not quite, and that's much to your advantage. Clawitzer pours most of its being into its Special Attack stat, which is brutally high. It bolsters this cannon-like propensity with Defense and Special Defense stats which are a bit above average, which, coupled with Clawitzer's few weaknesses, will allow it to survive at least one hit. The rest of Clawitzer's stats are pretty average, save for a rather puny Speed stat.
Mega Launcher: Pulse and Aura moves used by this pokémon receive a boost to damage. In essence, these moves become STAB moves... save for something like Water Pulse, which is instead doubled in strength. Yowza. It's your only choice for Clawitzer, and there's no reason why you wouldn't want this ability, given its moveset.
Clawitzer has a fairly large movepool, all things considered, but its singular ability normally leaves you with few choices as to what you should use. Your primary STAB weapon here is Water Pulse. Not only will it do damage on par with Surf, it has a decent chance of confusing your enemy. Very useful for an otherwise telltale pokémon like Clawitzer. Beyond that you have Dark Pulse, Dragon Pulse, and Aura Pulse, all of which benefit from Mega Launcher's boost. Clawitzer may be rather predictable, but in that predictability is a great deal of danger.
Should you consider other moves? Absolutely. Because Clawitzer has Mega Launcher, most players will assume that you're going to use either Water Pulse, Dark Pulse, Dragon Pulse, or Aura Pulse. Surprise them by using something else instead. For example, you could substitute in Ice Beam for Dragon Pulse to allow you to wipe out a great deal more pokémon at the expense of the extra damage, or swap Dark Pulse for Flash Cannon / Sludge Bomb to transform Clawitzer into an adept fairy-type killer. There are plenty of options open, though in all cases they will involve combat moves, not status ailments. Keep Water Pulse on your Clawitzer to add that element of uncertainty.
One last tip for Double and Triple Battle fans: Heal Pulse. This move doesn't get any use during Single Battles, but if you deploy a Clawitzer in the background during a multi-pokemon brawl you can use Heal Pulse to restore a massive 75% of the target's HP. Hoo boy, now that is ridiculous. (Just be careful not to accidentally use it on the bad guys.)
EV / Super Training
Clawitzer dishes out a lot of damage each round, so it behooves enemy pokémon to take it out on the first turn. Consequently, you'll want to work on its already decent defenses while EV training, particularly Special Defense. (Defense is important, too, but I find grass- and electric-type moves to be special more often than physical.) Shovel out roughly 180 EVs apiece for these, then dump the rest into Special Attack. Relegate more points to Special Attack if you plan on using moves besides the four Pulses.
Catching a Clawitzer
If you have Pokémon X catching a Clauncher, Clawitzer's predecessor, is fairly simple. Simply dip a Good Rod into the waters surrounding Ambrette Town, Route 8 or Cyllage City and you should come up with a Clauncher in a reasonable amount of time. Level to 37 to get a Clawitzer. You can also catch a Clawitzer in these same waters with a Super Rod, but the chances of landing one are pretty small.