The Northern Irishman, who left Swansea City for Anfield this summer, has taken the Reds through the third qualifying round of the Europa League courtesy of a two-leg win over FC Gomel and has acquired new signings Fabio Borini and Joe Allen so far in the transfer window.
"I do feel at home but I will never be complacent," said Rodgers. "I will come in and I'm sure the majority of supporters are still not sure about me.
"I've got so much to prove here. I have got to fight for my life to prove I am good enough to be here. I feel comfortable in relation to my position and work on the field. My promise was to fight for the club every single day."
Rodgers will take charge of his first Premier League game with the Reds when they travel to West Bromwich Albion on Saturday.