there are two basic ways to weaken a conclusion:
1 - undermine an assumption upon which the conclusion rests
2 - contradict an explicitly stated premise

it's extremely rare for a gmat problem to require an answer choice that does #2, since the truth of an argument's premises is usually sacrosanct. therefore, the vast majority of the time, you're going to be looking for answer choices that do #1: undermine an assumption upon which the argument rests.

here, then, are two strategies you can use for 'weaken' problems:
* 'forward' solution:
try to identify the assumptions upon which the argument rests. this is difficult at times, especially within the time frame allotted, but it's extremely helpful: if you can identify an assumption and then see an answer choice that knocks down that assumption, that's the right choice.
* 'backward' solution:
look at the answer choices, and ask yourself: 'what assumption would this answer choice be trying to contradict?' after identifying that assumption, think about whether it would be essential to the integrity of the argument. if so, you've got the right choice.

ironically, finding assumptions is even more important for these problems than for 'find the assumption' problems, because the answer choices themselves don't spell out the assumptions.