Oxford BCL extra-cirricularsWatch
Need more help on going postgrad?
Publications (including in student journals)
Research assistant work
Vacation Schemes (probably less relevant if you're into constitutional law)
Very few have all of the above (I certainly didn't and still got an offer)
Your best bet to strengthen your application would probably be to try and get some mini pupillages at public law sets, write some articles/case comments for the student journal/blog (if your uni has one), approach lecturers in relevent subjects to see if they could give you some research assistant experience etc.
Also, remember that references are a significant element of the application (probably more important that ECs) so make sure you spend your second year cultivating good relationships with your lecturers.