If the whole of a property is rented out the tenant is normally reponsible for the Council Tax. This would be specified in the tenancy agreement.
If you are just renting a room in your flat, the council tax for the whole flat remains your responsibility and you must pay it to the Council. It is up to you whether you charge your tenant a proportion of the tax in addition to the rent. But it is simplest to quote an all in figure covering rent, contribution to council tax, gas electricity and water. You need to agree with the tenant what happens for phone costs (unless you are each just using your own mobiles).