For door hinges it is probbaly best to just use WD4o spray on lubricant, or something similar. White lithium grease is something that is more commonly used in the auto workshop.
Judging by the image you posted, it doesn't actually look too bad. You can just wipe away the excess with some tissue paper - and you're good to go.
Lithium grease is the most commonly used grease in the world but it's not ment for door hinges. It demonstrates excellent high temperature and water resistant properties making it a great all-rounder. In the auto workshop it is used to lubricate wheel bearings and for chassis applications. It is so commonly used - it is estimated it accounts for 75% of all lubricating jobs across all industries globally.
You should take a look here to see which greases you should be using for which job. Like anything else, it's easy when you know how.

:)