It has been argued very convincingly that King Sejong himself was responsible for the invention of the script, and I think this is generally accepted by scholars today.

There were several writing systems that would have been known to Koreans in the 15th century other than those for Chinese and Japanese, such as the scripts for Mongol, Khitan, and Jurchen languages, as well as the Brahmic Siddhaṃ script for Sanskrit in Buddhist texts. Of these, the Mongolian scripts and the Siddhaṃ script were phonetic, and the Mongolian scripts were fully alphabetic. So the principle of alphabetic writing would not have been unknown.

Mongolian was written one of two ways, with the Old Uyghur script (as it still is today in Inner Mongolia) and with a script invented by the Tibetan monk 'Phagspa based loosely on Tibetan, which was used only briefly during the Yuan dynasty. Gary Ledyard has a theory that the latter was part of the inspiration for the shapes of some of the letters of hangul.

Sin Sukju (also romanized Sin Suk-chu) was a noted linguist and scholar under King Sejong. Starting in 1444, he visited the Liaodong Peninsula just across the border in China numerous times along with Seong Sammun (Sŏng Sam-mun) and Son Susan to consult the Ming phonetician Huang Zan who was in exile there. It should be noted that this was after hangul was first announced to have been invented, in 1443, and the purpose of the trip might have been more to study how to best represent the sounds of Chinese using the new alphabet (later, the scholars were to publish a dictionary of the Korean pronunciation of Chinese characters).

Well before Hangeul was created it was widely acknowledged that chinese pictograms were unable to represent the korean language properly. This was explicitly mentioned in the document promulgated by the government to introduce hangeul.

A system to represent sounds (which chinese pictograms can't do, they represent meaning) called "Idu" existed. It represented sounds by using pictograms that represented a word with a similar or equal sound, regardless of what the pictogram actually represents.

That probably doesn't make sense without an example so I'll draw an analogy in english:

• You want to write "How"
• The word "House" is similar (i.e. "Hou-se")
• So you draw a house to be read as "hou"

Similar systems to "Idu" had historically existed, for example the earliest example of japanese katakana is a buddhist text imported from korea (http://www.japantimes.co.jp/news/2002/04/04/national/katakana-system-may-be-korean-professor-says/)

What hangeul did was create symbols explicitly representing the various sounds but the principle already existed and was in wide use, particularly by the technical/craftmen classes.

Because they were focusing explicitly on symbols representing sounds the hangeul symbols are specifically based on the shapes made by the tongue or the mouth when making those sounds.