Characters pursuing this goal are desperate to glorify themselves, to make themselves known and be remembered by others. Because the means are secondary to the end result, even the most vile and reprehensible methods can be justified in their mind. Their sense of importance is bolstered by how many people are talking about them or what they're credited with (even if they're working under a pseudonym or alter ego).

Killing a vast number of people
Creating a high-profile, heinous problem so the character can swoop in and be the savior
Committing crimes that prove their prowess in an area (their intelligence, knowledge of a certain topic, ability to evade capture, etc.)
Publishing media (books, videos, a manifesto, etc.) that publicizes their beliefs and values
Starting a cult
Facilitating terrorist activities meant to frighten people or reinforce the character's ideology
Carrying out a large-scale event that culminates in their own death

There are five basic human needs that, when missing from a character's life, could motivate them to pursue this goal. The following needs are all possibilities, but only one of them should be the primary driver for any given character. For more information on the relationship between human needs and outer motivations, please see this Character Motivation tutorial.

Esteem and Recognition: A character with this goal feels as if they aren't valued or esteemed as highly as they should be. This is why they do what they do, because they believe that popularity or notoriety will prove their value to themselves and to others.