What do you mean access your character on website? If you're playing through web version simply login on main page of http://dragoneternity.com and press "Log-in" button that is near your character name you wish to play with.

I usually play on my phone.

Via the app.

like Ihit said just go to the website and login through the webpage

looks like you got online since your online now

I have access to my character through the app which is how I am on now but when I try to log on through the website it says that I don't exist.

Yeah, now the site finds me.