The trick is to use useradd instead of its interactive wrapper adduser.
I usually create users with:
RUN useradd -ms /bin/bash newuser
which creates a home directory for the user and ensures that bash is the default shell.
You can then add:
USER newuser
WORKDIR /home/newuser
to your dockerfile. Every command afterwards as well as interactive sessions will be executed as user newuser:
docker run -t -i image
newuser@131b7ad86360:~$
You might have to give newuser the permissions to execute the programs you intend to run before invoking the user command.
Using non-privileged users inside containers is a good idea for security reasons. It also has a few drawbacks. Most importantly, people deriving images from your image will have to switch back to root before they can execute commands with superuser privileges.