You can refer to the GitHub page "Duplicating a repository"

It uses:

• git clone --mirror: to clone every references (commits, tags, branches)
• git push --mirror: to push everything

That would give:

git clone --mirror https://bitbucket.org/exampleuser/repository-to-mirror.git
# Make a bare mirrored clone of the repository

cd repository-to-mirror.git
git remote set-url --push origin https://github.com/exampleuser/mirrored
# Set the push location to your mirror

git push --mirror

As Noted in the comments by L S:

• it is easier to use the Import Code feature from GitHub described by MarMass.
  See https://github.com/new/import
• Unless... your repo includes a large file: the problem is, the import tool will fail without a clear error message. Only GitHub Support would be able to diagnose what happened.