Build/Test/Deploy with Travis/Docker

Advice required; I'm looking into build/test/deployment solutions and have a github/travis account. I wish to run all tests on commit. We deploy docker containers and I'm not sure how best to run any tests. Should I build/run the docker container from a Dockerfile and then run the tests from inside the container? How would travis know if the tests fail when they are being run inside a docker container?

Any advice/experience is appreciated.