To do that, you need to first access your profile. You can do that in a number of ways:
- In the top bar, you can see your avatar next to your current reputation and your badge count. That is a link to your profile.
- Anywhere on the site where you can see your name — under answers, questions or on the question page — your username is also a link to your profile.
You need to do this from the main site (i.e. not from meta) to be able to advance to the following step.
Once there, note the bold, blue text reading Activity. This is your personal greeting-page when you access your profile. Next to activity are two links, one named Profile (this is what everybody else sees when they access your profile) and the other named Edit Profile & Settings. (If you’re still on meta it will just read Settings. Click on the small main user icon to its right.)
You will now access the editing page as can be seen below. From here on it is self-explanatory: make the corresponding changes and save.
Actually, there is one further thing. You are given two options of saving. One will apply the changes to all your Stack Exchange profiles. The other will apply them only to Chemistry (and Chemistry Meta). Choose wisely.
Finally, note that every time you change your username, you cannot re-change it for 30 days (except for a 15 minute grace period at the beginning). Meta.SE source