Basically like you always do, by adding it to the user’s .bashrc file:
FROM foo
RUN echo 'alias hi="echo hello"' >> ~/.bashrc
As usual this will only work for interactive shells:
docker build -t test .
docker run -it --rm --entrypoint /bin/bash test hi
/bin/bash: hi: No such file or directory
docker run -it --rm test bash
$ hi
hello
For non-interactive shells you should create a small script and put it in your path, i.e.:
RUN echo -e '#!/bin/bash\necho hello' > /usr/bin/hi && \
chmod +x /usr/bin/hi
If your alias uses parameters (ie. hi Jim -> hello Jim), just add "$@":
RUN echo -e '#!/bin/bash\necho hello "$@"' > /usr/bin/hi && \
chmod +x /usr/bin/hi