Burgundy as a political entity tooks many shapes throughout both Antiquity and the middle ages, with 3 kingdoms and 2 duchies, spanning a time from the last years of the Western Roman Empire to the XVth century, it rised before diseappering several times during this thousand years long period of time.

The Burgundy you are refering to is only it's latest incarnation. It originates as a powerful duchy within the Kingdom of France, it's dukes were among the most powerful and prestigious in the kingdom, being a cadet branch of the ruling Valois dynasty.

During the 100 years war, the duke Jean Sans Peur (John the Fearless) was assassinated by men under the orders of the Dauphin, the heir to the kingdom of France. This pushed Burgundy into an alliance with the English, helping them greatly in their victories in the later half of the 100 years war.

Using the weakening of central authority within the kingdom, the Burgundians continued to expand their influence as they have been for decades even before this. By marriage, money or conquests, the Burgundians added many territories to their domains, most notably in the low countries. The power vaccum created by the war only helping with their expansion which had already began beforehand. For example Jean had married the daughter of the count of Holland, which his son Philippe III le Bon (the Good) inherited. Philippe himself used the war to have a free hand he was able to expand further, for example by purchasing the marquisat of Namur in 1429 or by conquering the county of Hainaut in 1432.

The important date here is 1435, at that point the wind is turning in the 100 years war with the English more and more on the loosing side. The King of France, Charles VII, willing to severe the alliance between England and Burgundy, vital to the English, joined Philippe in Arras for a congress. There they decided to make peace, the Burgundian ending his alliance with the English and recognising Charles as king of France. In exchange, Charles promised to punish Jean's murderers and also lifted the Burgundian dukes from the obligation to pay hommage to the king of France.

This last part is the most interesting, the hommage ceremony is when a vassal swears his fealty under his suzerain, and a corner stone of feodal life. By not having to swear hommage the dukes of Burgundy where technically not a vassal of the kings of France. So Burgundy is an independant duchy then ? Well it's more complicated.

The convention of Arras didn't mention detaching the duchy of Burgundy from the kingdom of France. Moreover, the duchy was supposedly an "apanage", a special type of feodal holding given to a brother of the king and only inheritable through direct agnatic inheritance. Meaning that if the duke didn't have a son, the duchy was to be returned to the king of France. In this case it was given to Phillipe II le Hardi (the Bold) by his father Jean II le Bon, king of France, for his heroism at the battle of Poitiers This point is disputed however, as it was used as a justification for the reincorporation of the duchy into France following the death of Charles the Bold. Adding to the complexity is the status of other Burgundian holdings in France, like the county of Reuters and the county of Flanders which aren't mentionned.

So as you can see the status of Burgundy was a hotly debated topic at the time. The duchy itself was still part of the kingdom of France, but his dukes wasn't a vassal of the king of France, a rather unique situation in the Middle Ages.

When it comes to the relation with the Holy Roman Empire it is important to distinguish the duchy of Burgundy and the Burgundian Estates. The Duchy of Burgundy was only one title of the duke, although their main one, their other territories where held by their respective titles. The Duke controlled Luxembourg as duke of Luxembourg, and as duke of Luxembourg was a vassal of the Emperor. This was the same thing for all of his titles with the HRE, even though he was himself more powerful than the Emperor.

The situation was quite complicated, the burgundian territories were a patchwork of of feudal domains of various sizes accumulated during the 100 years war. This made them very powerful, at their height the dukes of Burgundy were probably the most powerful ruler in western Europe. As such they were de facto an independant ruler.

But their long time goal was to become a de jure independant ruler, and to do this they needed a crown. Fortunately for them a title of "king of Burgundy" already exist. Indeed Burgundy's previous incarnation was a kingdom located in south eastern France, sometimes referred to as the Kingdom of Arles after it's capital. The kingdom itself had diseappered centuries ago, but it's title still existed and was held by no other than the Emperor.

Having the Emperor giving him this title was the main foreign policy goal of Charles I le Téméraire, son of of Philippe III, leading him to for example mary his daughter Marie to Emperor Maximilien. However the early demise of the duke and his lack of a male heir will see his dream unachieved as his realm in being was split between the king of France and the Holy Roman Emperor.