What are some marketing strategies that you have used in the past or that have been proven to work well to obtain your competitor's customers and gain more market share?
Lot of ways
1. Target their fans on facebook - if they are big (20k+) you should be able to select them as an interest
2. Bid on the same keywords as them on adwords search and/or display - this requires some technical chops but there are ways to find your competition's placements. If you can get there and present a better ad and/or outbid them, you can get the clicks they are used to getting.
3. Find where their customers are spending time online - another thing that will take some chops but you can find out what blogs, facebook pages, etc your competition's fans are looking at and get in front of them there. This is an indirect method.
Most of the other options you will have are some variation on that theme. Make sure you have a way to stay in front of them (like retargeting or email marketing) if you want to make your investment go as far as possible!