This problem has different solutions depending on how you want to play.
If you're just looking for the achievement, one solution to your problem is to do whatever missions you want, choosing the side of good, until you get relatively close to your level point. Then go into a town and steal (for minor karmic reduction) or lay waste to entire towns (for major karmic reduction). You can quickly lower your karma this way. Then, once you're neutral, go kill some random creatures until you level.
(If you're looking to steal, a good place to do it is at Dukov's place… he's got stuff everywhere.)
The "nice" thing about this system is that if you stay on the path of "good" the whole time, you can do this at each achievement point: be bad until you get to the right karmic level, level up, get your achievement, and then restore a save to get back to your good self. You can quickly get "good", "neutral", and "evil" achievements for all levels on one playthrough this way.
(I did this, because I don't really enjoy long-term playing as an evil character… but I do like my achievements!)
You could also do this as an evil or neutral character, but it's much easier to quickly drop from "good" down to your new goal, than to work your way up.
If you find that you're close to leveling, and you're at the right karmic level, and don't want to take a chance on changing your Karma by doing a story-type adventure, you can always go wandering looking for random bad people to take out. Going near the Washington Mall can get you some XP fairly quickly, and you can go inside the capitol building and clear out large areas without picking up any Karma, positive or negative.
Campaign progress is stored online. Swapping computers (or reinstalling) shouldn't affect that.
Now that I think about it, I'm not sure if you can save a game in the middle of a mission like you could in SC1, and if so, I'm not sure if that would be saved. But completed missions are completed on other computers.
Update:
I have noticed some things that just don't get saved. Like if you finish a mission, then just purchase armory upgrades, then quit. It doesn't always save the armory upgrades. When the campaign is continued, it picks back up immediately after finishing the previous level. But this happens even on the same computer, and isn't any worse if you reinstall StarCraft II
Blizzard on Starcraft 2 Save Games: Starcraft 2 Blog
Best Answer
At first, the credits available to you in the single player campaign is limited to that from missions alone. However, once you've fully maxed out either the zerg or protoss research trees, you gain the ability to trade any future research points into 10k worth of credits.
There have been mentions of a glitch with the way the game handles this: if you replay a mission you've already beaten and acquire protoss / zerg research points within that mission, it is possible to get the Moebius foundation to "purchase research samples" from you, even though you failed to increase your net points (you already got them once). In this way, you can accrue as many credits as you need, and can buy all the armory upgrades in a single playthrough.Update: This method has since been patched.
As of now, the best way to earn the armory achievements is to play the campaign multiple times, selecting different upgrades on each playthrough, as achievements are shared across all campaigns.