I have class imbalance problem and want to solve this using cost sensitive learning.
sample_weight and class_weight have a similar function, that is to make your estimator pay more attention to some samples.
Actual sample weights will be sample_weight * weights from class_weight.
This serves the same purpose as under/oversampling but the behavior is likely to be different: say you have an algorithm that randomly picks samples (like in random forests), it matters whether you oversampled or not.
To sum it up:
class_weight and sample_weight both do 2), option 2) is one way to handle class imbalance. I don't know of an universally recommended way, I would try 1), 2) and 1) + 2) on your specific problem to see what works best.