Well this is a little difficult, we cannot know the process of thinking in William's mind that led him to invade England for sure and many of the sources around these events are deeply biased.

So where did the idea come from? Its hardly likely that he just woke up one day and decided that conquering an island kingdom would be a fun thing to do after all! Norman propaganda claims that he was promised the throne by Edward the Confessor in the event that he died without an heir, and that this claim was supported by Harold Godwinson (and sworn on sacred relics, Harold supposedly denied the validity of this oath as it was sworn under duress). However the truth of this matter is somewhat murky. While it is not far fetched or totally impossible for Edward to favor Normans, Edward spent much of his life in exile in Normandy and he certainly had a reputation for favoring Normans within his court, (though this too might be the result of later propaganda) it is a bit of a stretch to imagine him fully promising William the succession, especially since Edward still had living male relatives (young as Edgar the Ætheling was). That is not to say it could not have happened as William claimed, just that we need to be skeptical and take other factors into account as well.

As for the invasion itself. This is a more fraught issue. In general medieval armies at this time period, and later, avoided pitched battles as much as possible. Instead sieges and raiding (chevauchée in the 100 years war might be a better example) were designed to bring about negotiations and resolutions. Staking everything on a pitched battle was incredibly risky, so leaders avoided it as much as possible. Battles were unpredictable and potentially devastating, sieges and raids were a lot easier to manage. William taking the fight to England and offering battle was incredibly bold, but also his only option really. As you correctly note, he was somewhat cut off from his base of support and supplies. While foraging the countryside would keep his army fed for a while, he was in hostile territory in autumn with winter, and the effective end of campaigning season in pre-modern times was on the horizon. He also could not stay too long in England for political reasons, the absence of a medieval ruler was usually taken as an invitation by his rivals for some meddling, and William's relations with his neighbors in Brittany was strained at the best of times.

Now the Battle of Hastings itself has been the topic of untold books and articles. The long and short of it is that William's gamble was extremely close run for most of the fight and his victory was far from guaranteed. He did indeed emerge victorious and perhaps under the best possible conditions. English leadership was devastated by the battle as the king and his immediate obvious successors, two of his brothers, were killed in the battle along with much of the "professional", for want of a better word, soldiers such as the huscarls. The rest of Harold's male relations fled abroad or were imprisoned by William.

But now we get to the hard part, winning a kingdom in battle is the easy part, but ruling it is a whole other matter! Indeed William had quite a bit of trouble in his first several years as English monarch. Revolts in the Northern England required his attention, as did Danish and Scottish raids (in support of Edward's remaining male relative Edgar or just trying to capitalize on English weakness), and his holdings in Normandy were not totally secure either. However through the process of the Harrying of the North (and the near desolation of Northern England if our sources are to be believed), Harold's sons causing trouble in Wales and Ireland, along with the breaking of English native nobility through the defeat of the Northern earls as well as other prominent rebels, and the eventual reconciliation of Edgar and William, the Norman Conquest was more or less solidified. Establishing control over England by the Normans was a long and bloody affair and it took decades for it to be fully completed. In its wake were not only the military dead from Hastings and revolts later but untold numbers of civilians killed through the engineering of a famine in Northern England as well as violence against civilian populations.

Other efforts also helped the process along. The English Church was quickly subsumed into the Norman administration, and William went to great lengths to portray his succession as legitimate, through both propaganda efforts such as having Edward the Confessor canonized as well as through a policy of letting sleeping dogs lie in many cases. William did not change the legal codes of England in the immediate aftermath of his conquest and seems content to have merely co-opted existing legal practices to smooth the transition.

So the tl;dr, William got extremely lucky at Hastings, the conquest itself took years and lots of pain and suffering that was aided through institutional support from the Church and the co-option of existing legal norms to smooth things over.

Partly as a 'joint stock enterprise'. He decisively (including such as getting personally endorsed by the Pope) heading the force and provided the core and largest contingent - Normandy estimated as supporting some 600 knights + ships, lesser troops and supports in proportion and large numbers of mercenaries - mainly but not only foot (though it not known where he got / raised the money for this). Which he had over several decades formed into a formidable and largely loyal and biddable force. But he also 'recruited' bands-companies-lords? (though no princes etc. who might compete with him) from Brittany and Flanders - each estimated at amounting in total to 300 or more knights + 'some French knights. Promising them plunder and a cut of any territorial gains.

He then very cleverly timed his crossing e.g. letting the Norwegians and Tostig (who he not only knew of but liaised with) invade first and a time of favorable winds and sea conditions (there are a few comments as to a 'good passage' and of course his seamen fully familiar with these, often difficult, waters).

His logistics 'seem' to have been quite sophisticated for the time (remembering that he a veteran campaigner - often far from home e.g. Maine etc.). His fleet having sufficient capacity and carrying horses, supplies and even prefabricated fortifications. While he of course supplied himself as far as possible by plunder-grabbing everything he could when in England.

All in all he showed himself to be a superior - perhaps even the best contemporary commander in Western Europe.