Once married, a step parent has the same legal rights to claim a dependent as the biological parent, even if a formal adoption has not taken place. You can claim your children as dependents following all the normal rules (must be US citizens or residents, must have social security number, must live in your home more than half the year, etc.). The rules are in publication 501 if you want to see them in writing, but there is nothing special or complicated about your situation.
https://www.irs.gov/pub/irs-pdf/p501.pdf
