My key was password protected which was causing the problem, a working file is now listed below (for help of future googlers)

FROM ubuntu

MAINTAINER Luke Crooks "luke@pumalo.org"

# Update aptitude with new repo
RUN apt-get update

# Install software 
RUN apt-get install -y git
# Make ssh dir
RUN mkdir /root/.ssh/

# Copy over private key, and set permissions
ADD id_rsa /root/.ssh/id_rsa

# Create known_hosts
RUN touch /root/.ssh/known_hosts
# Add bitbuckets key
RUN ssh-keyscan bitbucket.org >> /root/.ssh/known_hosts

# Clone the conf files into the docker container
RUN git clone git@bitbucket.org:User/repo.git