So After William conquer England, he kept his Norman lands, was he still a vassal of the French King as the Duke of Normandy or was Normandy now a part of England? Did he have to pay tribute to the French King in any way?
The Duchy of Normandy remained a de jure part of the kingdom of France even after the Conquest. We know that William often fought with other French dukes and the French king Philip I, before and after the Conquest. I do not recall any instance in both primary sources or secondary sources where William gave homage to Philip I (which does not mean he did not but simply means I have not found any, mea culpa), but had he done homage he would have done so in the vestiges of the duke of Normandy, not as the king of England.
De Facto the duchy of Normandy was part of the kingdom of England. Taxes, men and loyalty were due to the duke of Normandy, who was the king of England. There were several legal problems, since many nobles in Normandy had the right to appeal decisions made by English kings to French kings. This was normal in medieval societies and many of William's successors did homage to the king of France recognizing his lordship over their lands. In turn, French kings would recognise the rule of English monarchs on French lands. This system was maintained, but it put French monarchs at odds with English ones. The French monarchs either wanted lands back or desired to end English influence. English monarchs on the other hand wanted an end to French over-lordship and their legal influence in their French lands. Ultimately culminating in the Hundred Years war.
To summarise, William the Conqueror remained a vassal of France as the duke of Normandy and owed the king of France homage for his duchy. As king of England, however, William was equal to Philip. Hope this helps!