PART 1: It was almost accidental. At no point when he began uniting the Mongol tribes did Chinggis Khan (the more accurate rendition of Genghis Khan) seek out world domination. Rather, it sort of kept happening.

What people don't realize is that Chinggis Khan didn't succeed in uniting the Mongols until he was about 50, declaring the Mongol Empire in 1206 (he was born in 1162 as Temujin, and it wasn't until 1206 when he began being called Chinggis). That was many decades of slow warfare and his own losses and defeats before he succeeded in doing so. Initially he was probably concerned only in securing his own position and safety, perhaps at best hoping for leadership of his own tribe. Events went differently, and a 'concerned citizens' alliance of steppe notables unhappy with Temujin's rise (such as appointing his generals and chief lieutenants from common folk based on their ability, rather than old bloodlines) turned the conflict into a massive civil war for control of Mongolia. By this time this was complete, Temujin had restructure Mongolian society, breaking the power of tribal Khans and old tribal ties, placing himself, the Great Khan, at the head, with discipline and utter loyalty the byword, harnessing the great military potential of the steppe horse archer.

By 1206 then he had a fierce military force, a great spear: but had almost run out of enemies in Mongolia itself. All dressed up and nowhere to go, you might say. What Chinggis Khan understood was that he needed a common enemy, or old internal intrigues would rise back up to the surface and tear his new union apart. The spear needed to be thrown, lest it fall apart. The obvious answer lay to the south, the kingdoms of northern China, the Tangut Xi Xia in modern Gansu/Ningxia, and the Jurchen Jin Dynasty controlling from Manchuria to the Huai River, capital at Zhongdu (modern Beijing). Whether Chinggis at this stage was intending to conquer them, or attack and raid is hard to say. Personally, his actions to me suggest the latter, but some argue for the former. There certainly were pretexts to attack both. The Tangut had allowed certain enemies of Chinggis to flee through their territory or seek shelter, and the Jin had long been enemies of the Mongols, breaking up an earlier Mongol union (commonly called the Khamag Mongol confederation) in the early 12th century. While there certainly was this pretext, it should also be noted that a quite dry period in late 12th century Mongolia, and continuous warfare, had depleted the herds the Mongols relied on to survive as well as cost them in other goods. The raids, especially against the Xi Xia, may have just been out of need to replenish these stocks. Part of this too may have been encouraged by both states entering into a period of instability and poor emperors: as good an opportunity as any for the Mongols to attack. If they had aspirations beyond this, I don't think we can say, and it seems highly unlikely to me. Chinggis Khan's intentions here seem very localized.

In 1209 the Mongols invaded the Tangut Kingdom, making off with a great quantity of goods and forcing the Tangut King to submit to them (which entailed him sending a massive amount of tribute and slaves). Not long after his return from the Xi Xia, the Jin Emperor demanded the submission of Chinggis Khan (Temujin had undertaken a formal vassalization to the Jin in the past, and the Jin may have been surprised by his sudden unification of the steppes and understandably worried, but were distracted by war with the Chinese Song Dynasty to the south), who insulted the envoys and used this as his official declaration of war, invading the Jin Empire in 1211.

I think what turned Chinggis' mind to more permanent conquest than he initially intended was the success the Mongols had. While sieges were difficult, but aided by Jin defectors providing them siege weapons and knowledge, Chinggis Khan's army absolutely devastated in almost every field encounter. Though the Jin had mighty horsemen and huge numbers, they were unable to gain local superiority of forces and suffered from defections after defections. Within a few years, there were more Chinese fighting for the Mongols against the Jin that there were Mongols there!

By 1215, Zhongdu had fallen (with some difficulty), and forced the Jin Emperor to supply tribute and Chinggis withdrew back to Mongolia, but when the Emperor soon broke this agreement by fleeing to Kaifeng, war resumed. Was Chinggis intending a conquest of all of China now? Well, still it is not quite clear whether his intentions were to force the Jin to submit, fully conquer them or take all of China. I personally have seen no evidence he desired all of China yet, and the Jin were still a major obstacle. I think his intentions did not go much beyond forcing the Jin to be his vassal.