Firstly, you should place your BurpSuite jar file to the root of the cloned repo and change BURP_VERSION in Dockerfile accordingly.
The folders share, .java and .BurpSuite must be owned by a non-root user before you start container, otherwise you may lose your license activation, so use carefully.
If you already have an activated license, just copy it to .java or specify volume from your $HOME/.java to /home/burp/.java in docker-compose.yml and you will probably be fine (NOT TESTED!).
Run Burp:
sudo docker-compose up -d
By default Burp proxy will listen on the local interface only, preventing docker from exposing ports to host properly. So you should change it in Burp settings during first start to all interfaces. After that the local port :8080 on your host should work as expected.