First make sure the new GFCI is wired correctly. Look at the GFCI. The hot and neutral screws are in different places on different manufacturers' outlets. Connecting them wrong will ruin the new GFCI. This happened to me when I had a neighbor help with some wiring. He wired it just as he wired his - but mine was different and ruined the GFCI.

Its also important to connect the hot lead to the line screws and the wires leading to the other circuits to the load screws.
Does the breaker keep tripping?

There are other outlets on this same circuit. They may be in different rooms or even in the basement. Is there a heater or other appliance plugged into one of them?
You are either drawing too much current on one of them or one is shorting out.

Also, is the new GFCI and breaker of the same amperage as what they replaced?
Finally, it is possible the new GFCI is broken.