Most accounts are paid monthly or yearly via the Stripe credit card handling service and you need take no action to renew, provided the card you used to sign up is still valid.
If you want to upgrade your account (for example from a free trial or to a more advanced version than you currently have) you can do do using the System Menu (click your name at the top right of the screen) and choosing the menu item “Renew or upgrade”. This is how you always need to complete your free trial if you wish to continue with Artlook.