Gather any documentation that verifies your financial situation has improved since you applied with the help of your cosigner. Examples include bank statements, your credit report, pay stubs and similar data.
Contact your cosigner and discuss removing their name from your lease if he does not already know of your intent. You must do this, as your cosigner legally is attached to your rental agreement.
Call your landlord and, working with your cosigner, make an appointment to discuss your lease where all parties can be present.
Go to your lease appointment with your cosigner, dressed professionally. Present your documentation to the landlord or property manager and ask if she will consider letting your cosigner out of the lease. If your landlord is very understanding, she may draw up an addendum concordant with your request, releasing your cosigner from his obligation.
Ask the landlord if she will consider running a new application for you to see if you qualify for a lease on your own. If she agrees, fill out any paperwork the landlord or property requires of you in order to run a new application -- you may have to pay the standard application fee for all tenants even though you already have a lease, as the application fee covers the cost of your new credit and background check.
Sign the addendum or new lease that doesn't have your cosigner named. Request that the landlord or property manager make copies of the addendum or new lease for you and the cosigner so everyone has proof that the cosigner no longer is attached to the rental property through you.