Will low community college gpa keep me out of grad school?

I went to a community college for about five years, and the first four semester were rough for me. I didn’t get stellar grades, and even got an F in a biology class. I pulled it together, and left with a 2.5 gpa. I transferred to a well renowned state school and so far have a 3.6 gpa. I figure I can leave with at least a 3.8 but I am worried how grad admissions boards will look at my Community college grades. I don’t necessarily want to go to any Ivy league schools (though that’d be nice) I just want a decent chance to get into a respectable program in either public/foreign policy. Will my low CC gpa hold me down? if I go back to CC and take some more classes can that help my gpa?