Before we get to being an ulzzang, we've got to break down what an ulzzang is. 

Ulzzang is a romanisation of the korean term 얼짱. 

얼 comes from the korean word for face, 얼굴. While 짱 is the korean slang for best, or amazing.

Put them together, and you get the literal translation of ulzzang - best face. 

Though it literally refers to anyone with a pretty face, due to it's rise in population in mid 2012, it has developed into a culture. Now instead of being isolated to just South Koreans, it has spread to surrounding Asia countries. On social media platform, namely Tumblr, there is a rise in non-asian people striving for the title of an ulzzang, though it is to mediocre success. It is important for me to emphasis that the standards of "pretty" or "handsome" is based off standards of beauty in Korea, thus most traits are attributed to Asians. Do not confuse it with a culture that ostracizes people of colour! It is just that as a culture bloomed out of Asia, it's 'traditions' are made to apply only to Asians. Personally, I feel that ulzzangs can be from people that aren't Asian, though they may not be recognised as ulzzangs by some people. Although the term ulzzang refers to good looking people, it has a really strict set of standards. 

For females, this usually refer to light-skinned, thin women with an unblemished face.

For males, this usually refer to muscular / thin men with an unblemished face.

It is important to note that the face is the most important thing, one's weight doesn't really matter. However, Koreans do have a strict culture of not being fat. We've seen an evolution from (sickly) thin to healthy looking people but even then, an ulzzang usually tend to be on the small frame of body (even the males).

Anyone can be good looking, so ulzzang culture defines an ulzzang by someone who is recognised by the public. This means that you'll have to have a substantial amount of following on social media platforms - usually Instagram or WeiBo. It is important to understand that this term doesn't equates to model. Unlike the west, where Instagram models would put the word "model" somewhere on their bios, ulzzang is a net term. It is used by the public, instead of the person itself. In a sense, you can't refer to yourself as an ulzzang, because the term has to be given to you by others. Ulzzang, as a word, is used to describe another, and not oneself.

Understandably, as a culture derived from one's looks, it is harsh and quick to throw shallow judgement. If you want to be an ulzzang, it is best that you prepare yourself for judgement. Good looks are flexible in definition, hence one's appearance can please one and irritate another.  

TLDR ; ulzzang refers to good looking people, with the standards set by Korea.