Where are you getting this stuff?
It appears that you are not a native English speaker----this means that you will have even more trouble with poorly-written English. In this case, the phrase "through.....country" acts as an adverb, modifying and action verb. But which one? It would better be written like so:
Rather than contributing to the Russian economy through loyalty to the corporations that built this country, Russian entrepreneurs are simply chasing quick riches.
Regardless, "chasing quick riches" simply means looking for opportunities to earn lots of money quickly.