First, assuming that you are a leaseholder, check your lease for any conditions relating to adding further structures, specifically a balcony. Next, ask the freeholders if they will apply any charges or fees if you erect the balcony, should planning and building regulation approval be granted. You will need party wall agreements with the other owners/freeholders etc if you do not own the external walls that you will use to secure the balcony.
A flat does not have Permitted Development rights and so you will need to apply for full planning permission for the proposed balcony. The planning application will consider the impact on the building, the neighbourhood and the adjoining owners. Look at previous planning applications for balconies in your area to see if there have been any approvals.
You will also need a building control approval. The building control inspector will consider the structural design of the balcony.