Milking the system means to obtain money or benefits from it (as in to obtain the greatest possible advantage from). While bilking the system means to swindle or cheat (ex: made millions bilking wealthy clients on art sales).
Article references to "milking" and "bilking" the system:
The real benefit cheats are the employers who are milking the system. - The Guardian
MIGRANTS were accused of “milking the system” after figures showed they are more likely to claim benefits than people born in the UK. - Express.co.uk
A $712 million bust, the biggest in U.S. history, shows that the people most likely to bilk the system are doctors and medical providers, not “welfare queens.” - The Atlantic
An owner of a defunct company accused of bilking the Defense Department out of more than $20 million, including charging nearly $1 million to ship two 19-cent washers, pleaded guilty Thursday to federal wire fraud and money laundering. - CBSNews.com