1. Login into your twitter account
2. goto -> settings(on top near to search, in the drop down last but first option, just before sign out) -> account(on leftside menu first option)
3. You can see your full account details like username, time zone, mailid etc.. In username textbox type your new desired username and wait for twitter's response regarding the availability of that username. Try with some modification in the username until you get username available message. Once you got go to the bottom of the page and save changes.
Thats all now your username is changed to new one as you entered.
Note:
1. Details are based on the look of twitter in desktop/laptop web browser not from mobile browser.
2. Once your username is changed you should use that new username while signin into your twitter account.