First of all, make sure your favorite bash is in $PATH before the system one. Also set the environment variable SHELL to the full path to your favorite interactive shell (presumably your own installed version of bash).
You can't set your own shell as login shell without root's intervention, but usually you don't need anything better than POSIX sh in ~/.profile anyway. You can put exec $SHELL at the end of ~/.profile so that you get your favorite interactive shell in console or ssh sessions, but make sure that you don't do it from shells that are not interactive (for example, some Linux distributions have X session scripts that start with #!/bin/bash --login). It is difficult to detect precisely when it is reasonable to call exec in .profile, but testing that the shell is interactive works in most cases; you can do something like this:
case $- in
*i*) :;;
*) exec "$SHELL";;
esac
If you're not root, that's about all you can do. In particular, scripts headed #!/bin/bash will keep on using the system bash, while scripts headed #!/usr/bin/env bash will use yours.
If you are root, you can add your bash installation to /etc/shells, after which you can use chsh to change your login shell to it. You can also replace /bin/bash by your own version, but I wouldn't recommend this for such a critical system component. If you really want to replace /bin/bash, at least keep your package manager happy; on a dpkg-based system, you can use dpkg-divert.
