BURP depends on BOINC being installed.
Well, it is a little complicated actually:
1) You start out by installing the BOINC backend (the server, not the client). Once you've got that up and running and tested you can move on to (2)
2) Then you run 'sh ./setup /path/to/project/base' from the checked out BURP code. This installs the BURP web-frontend on top of the default BOINC one.
3) Next you need the BURP backend server (also called the BURP Core) in order to actually see anything useful in the frontend pages. To get that you must install a Tomcat/Glassfish/Jetty or similar server for Java-based web containers. Then compile BURP and install the .war file
4) Go to the core backend URL on your server and type in the database credentials, file paths etc. for BOINC (if they are not detected automatically). The DB connection is something like "jdbc:mysql://:3306/yourproject" but each setting in the Core is pretty well documented when you hit edit for it, so just follow the instructions there.
4b) Once done with the database setup, restart the container or hit "Retry / Reconnect" somewhere on that page if the option is available.
5) BURP Core will unpack the required additional database tables and install them in the database for you alongside the existing BOINC ones. You will be able to see this happen in the server log file as each feature is enabled one by one on the very first run (and additional changes later if you update from one version of BURP to another at some point in the future).
Then you should pretty much have an empty version of the current BURP website.
If you are interested in CSS too then you should be aware that there is not just one, but two different frontend looks checked into the code repository (set in /burpproject/header). The one currently used is the one called "old". The whole thing is auto-generated using SCSS with Compass extensions.