What is the easiest way to complete "Kill x number of enemies" missions in Helldivers?

If you mean the arena missions, where you just need to kill a certain number of enemies, they are actually very easy if you got the right set of stratagems. I assume higher difficulties in my answer; easy to medium difficulties might allow some other strategies that won't work on Helldive and above (12+).

EXO spam

Each of you takes 4 EXOs. The EXOs you take need to be fully upgraded, or you will run out of ammo quickly.

• EXO-44 "Walker" (the one with a minigun and 8 rockets) is rather viable against Bugs and Illuminates (granted that you can reliably hit the centre of enemy units), but its anti-tank potential is rather limited, so it doesn't work that well against Cyborgs. You also need to unlock it, which is not always possible. There is not a lot of ammo in its minigun, so make your shots count.
• EXO-48 "Obsidian" (the one with two 20-mm cannons) is just awesome against Illuminates -- you have a lot of ammo, there are no enemies with permanent tank armour that you cannot breach, its damage and rate of fire are very high, etc. You cut through Illuminates like a knife through butter. Any difficulty is very easy with 8 fully graded EXO-48s. However, when it comes to Bugs, you are in trouble once you encounter Behemoths. You are in lesser trouble when you see Impalers and Tanks, but it is still a trouble. And when it comes to Cyborgs, you are in a huge trouble, since there are really a lot of units that just flat ignore anything but anti-tank weapons. Overall, you need some additional anti-tank support when you go against Bugs, and a lot of anti-tank "support" when it comes to Cyborgs.
• EXO-51 "Lumberer" (the one with an anti-tank gun and flamethrower) has some problems. You cannot refill ammo, and it is very limited, much more than it's with other EXOs, so you will run out of ammo rather quickly if you don't conserve it. Most of the time, you cannot afford to waste main cannon ammo on non-heavy units. The other problem is that you will damage yourself if a heavy unit comes too close -- and all EXOs are very, very slow, so you will need your friend to shoot the enemy from another direction. However, this EXO performs rather well against Cyborgs, the race that basically spams heavy units, especially on high difficulties.

If going for EXO spam, it might be a good idea to take one REP-80, a repair tool, but if you are careful enough, you won't need it, as you will run out of ammo before your EXO is significantly damaged.

Tank spam

TD-110 "Bastion", if fully upgraded, has a pretty large magazine, and, what is more important, is issued with a machine gun. It is very fun to use if you can coordinate well. The main benefit is that most units cannot even harm you at all.

Some tips on using it:

• Use machine gun (alt-fire/aim button, right mouse button on PC by default) to kill light- and medium-armoured units. Always aim for the centre.
• Turning is quicker if you do it around one point, without going forwards or backwards.
• Before firing the main gun, "prefire" with a short MG burst to see where the bullets go. The gun shell will hit exactly there. This helps you to avoid missing the target and hence wasting ammo, and, what is even more dangerous, harming yourself by firing at someone too close to you.
• Again, be very careful, don't fire it when you are too close to an enemy!
• When possible, don't waste MG ammo (not to mention the main gun shells!) on units that you can just run over. As a bonus, it is more fun than using your MG!
• Communicate about what are you doing, though. From my experience, it is better when the one at the driving position is in charge and tells when to fire.
• Always move! Even though your tank is almost indestructible, almost doesn't mean completely. Some enemies can still kill you very quickly. Especially beware the Illuminate Obelisks that can summon walls. If this wall hits you, you are one-shot-killed.
• If someone needs to leave the tank for a moment, it is better for the gunner to do it.
• Don't go backwards as the driver when the gunner is going to leave the tank, and don't leave it as a gunner before telling that you gonna do it. Don't exit the tank as a driver after going forwards before it is fully stopped. If you do any of those two things, you are likely to be smashed by it.
• It is a very good idea to bring a REP-80 repair tool with you, as tanks tend to get damaged before they use all the ammo. If you don't have it, don't hesitate to leave a tank that has lost half its health.
• A Displacement Field perk pairs very well with a tank, allowing you to exit it at the last moment when it starts to burn.

Things to consider:

• Just as with Obsidian and Lumberer EXOs, it is a DLC thingie, so if you don't have DLCs, bad for you. On the other hand, there is no need to unlock it, and for 6 upgrade points, you can get a very potent killer machine.
• If your tank is destroyed while both of you are inside, the mission is lost. If you are one by one, you might have time to dial a Reinforce Statagem.
• You cannot summon stratagems from a tank, but you can when on foot and from an EXO.

Mine spam

Air-dropped anti-personnel mines can damage tanks when upgraded to Antitank mines (3d level), and you have 10 uses of them. Any unit stepping on them, including Bug Tanks and Cyborg IFVs, is instantly killed, if not by the first mine, then by another one, you can actually easily make a field of them. The best perk to take is Stratagem Priority, lowering already not so big KD of 45 sec to 27 sec. Just find a tight corner and surround everything with mines! Be careful, though:

• Don't roam around with the stratagem dialled in your hands. If you are knocked out, you will summon mines on yourself, possibly killing you and your friend(s). Only dial them where it's safe to drop them.
• Watch where you drop your Reinforce stratagem. Don't drop your friend(s) on a minefield.
• It is not a bad idea to have a "plan B" for enemies that somehow pass around the mines.
• This strategy pairs perfectly with LIFT-850 Jump Pack -- you cover the map with mines, you jump around, enemies go bang-bang on the mines. However, you might need Resupply stratagem in this case, as your jetpack will run out of fuel. You will need to jump a lot more on such missions.
• To save up some mines, you might want to shoot weak enemies passing on them. This way you will save the mines for heavy enemies.

AT-47 Anti-Tank Emplacement

The Anti-Tank Emplacement is the same 90-mm anti-tank gun, but stationary, and:

• Is very resilient.
• Has a lot of ammo.
• Only needs one person to fire it.

As of the "Dive Harder" update, they can harm themselves, though.

The best perk to use with AT-47 is Stratagem Priority. It is really a good idea to bring a REP-80 rep kit: otherwise, the turret gets destroyed before you have a chance to expend all of the ammo.

You should place your turrets so that they are relatively close to one another, but not close enough to damage one another by shooting enemies at point-blank range. Each player should have their own sector to cover.

This pairs very well with Static Field Conductors, which slow enemies down. It is better to make corridors for enemies to cross, so that you don't have that much space to cover.

Tesla spam

A/AC-6 Tesla Tower, when upgraded, fires 3 bolts per attack, being very effective at clearing out hordes of enemies. As far as I know, it is not anti-tank, though, so it is only viable against Illuminates and on very low difficulties, which I don't focus on in this guide. Be careful when placing them:

• Use Stratagem Priority to summon them more often.
• Leave no space between turret ranges.
• However, don't let the turrets be in the range of one another -- they cannot distinguish friends from foes. This is about you too: don't ever get in their range, at least not when stood up.
• Choose tight corners, so you can cover everything around you with them.
• REP-80 is very helpful, as enemies actually damage them.
• If you fight someone who also has tank units, it is definitely not a bad idea to bring anti-tank weapons. Or choose another strategy to use.
• Don't summon your fallen friends in the Tesla range, just to fall again...

The biggest problem is that it is a DLC item, you need to buy it. No need to unlock, though.

Minigun turret spam against Illuminates

Because why not? They can penetrate anything against that race. Their RoF and DPS are very high. You can heal the turret with REP80, go prone, and just watch the mayhem.

Since you get the turrets from the start, no need to unlock or buy them.

You can also set the turret up far away, so even if it hits you a bit while aiming at some enemy, it is really just a bit.

Normal setup against a given race

Those were all "cheap" solutions, your primary weapon wasn't listed most of the time because of no need in it. But you can also learn to play this game well enough, take normal equips against your enemy race, and just play it normally. For example, use Recoilless Rifle against Cyborgs, take Resupply stratagem, run around, kill enemies etc. This is harder, but might be more fun.

Good luck!