We don't generally underline or quote to mark a person's name. However when we want to emphasize it, it can be underlined, quoted, or marked by dots above the characters, like .
Refer to the Korean Grammar Rules(한글 맞춤법) by the National Institute of Korean Language:
Supplement 18. Emphasis marks (드러냄표; ˙ ) and underline ( ___ )
Used to indicate the notable or important parts of your sentence.
한글의 본디 이름은 이다.
가 아니라 [!]
지금 필요한 것은 _지식_이 아니라 _실천_입니다.
(I cannot underline in Stack Exchange, so think of text between _ to be underlined.)
Instead of emphasis marks and underlines, single quotation marks can be used.
한글의 본디 이름은 '훈민정음'이다.
중요한 것은 '왜 사느냐'가 아니라 '어떻게 사느냐'이다.
지금 필요한 것은 '지식'이 아니라 '실천'입니다.
I think almost no one uses emphasis marks to emphasize nowadays; most people use underlines, bold characters(굵은체 or 볼드체), 'quotes', or sometimes oblique characters(이탤릭체). I, however, am against the use of oblique characters in written Korean since it is hard to read.
There is no special way to distinguish Hanja words(한자어), pure Korean words(고유어, 순우리말), and foreign words(외래어). Even Koreans are confused of this.
For example, some Koreans confuse 생각, 마진(margin), 가십(gossip), 댐(dam) to be Hanja words. They may appear to be when written in Hangul, but they are not!
There are more words that are thought to be traditional Korean words, but are actually foreign words, such as 가방 (from Japanese Kaban
かばん), 담배(tobacco) and so on!
(Actually we can tell if a word is Hanja, foreign or pure in most situations, since the words from each groups share common pronunciations. We have a sort of intuition that words like sushi and tempura are Japanese words and kimchi and bulgogi are Korean words, right?)