They are famed for their white eyes, which hold an incredible power. They can use their dojutsu to see the chakra network of others and they make exceptional sensors because of this Byakugan.
Their ideology follows a strict traditional footpath that has carried with them throughout generations, it is to show respect, and above all else show the righteous strength of the Hyūga clan. They absolutely revere the "Honour" system and code that bounds them and find themselves to be in a higher regard than most other Shinobi as they are among the "elite" of Konohagakure and as of such are supposed to show it like that.
Wordlessly saying, they are loyal to themselves and Konohagakure, though they do not find themselves any friends in the surrounding clans, they make do with the acquaintances. The only other significant relation is with the Uchiha clan, seen as their innate rivals and have always been somewhat distasteful with one another; though not to the same extent as the Senju lineage.
Upon their personality they bore the trait of pride, and find their Taijutsu to essentially be "flawless" and the only way their Taijutsu is flawless is if the user is flawed. They find a lot of pride with their members reaching their highest potential, even though it is not emotionally expressed.
The clan has manifested through Hi no Kuni as among the highest and most notorious ranking members which were coveted for their abilities. There hasn't been a time in which the Byakugan hasn't been noted, the 'lavender' esque eyes which are help their namesake. Like when the Senju clan and the Uchiha clan had melded their abilities together to reign over the country. The Hyūga clan were not present during this, but were more than happy in the creation of the hidden villages.
The Hyūga most notably came to light during the long lasting shinobi war twenty-four years ago. They were coveted for their abilities of assassination, and brute force of decimating enemy troops. Before that time, they were still known as a noble and founding clan of Konohagakure; traditionally seen as "Guardians" during their construction, overseeing the transportation of materials, missions during the crisis and even helping in the building. Now, they are considered among as the elite clans of Konohagakure no Sato and are held in high regard.
They currently oversee the more political side of Konohagakure and have deep roots within it's decision making, alongside this, they are training up their young to take over the next generation, to upkeep tradition strict and their bloodline strong.
As seen in the image, the clan is separated into two parts, the Main House (宗家, Sōke) and the Branch House (分家, Bunke), a system that protects the secrets of the Byakugan from outsiders. The Main House runs the family while the Branch House protects it. The members of the Branch House are branded with a cursed seal upon the third birthday of the heir to the Main House.
This seal gives the Main House members absolute control over the branch members, because they can destroy the branch members' brain cells with a simple hand seal, or cause them great pain to punish them. This seal also makes sure the secret of the Hyūga's Byakugan is safe, because it seals the Byakugan after the wearer of the seal dies. Because they are essentially servants, simply because of their birth order or heritage, some Branch House members are hostile to those of the Main House, though not directly.
It should be noted, that only the Main house members are allowed to learn the Gentle Fist techniques. The Bunke branch, outside of the Gentle Fist style are forbidden to learn any other techniques that are exclusive to the Sōke branch members.
The Byakugan ["White Eye"], is their bloodline. It is a Dōjutsu. The first of abilities is when activated, the Byakugan gives a near 360º vision, except for a small blindspot at their first thoracic vertebrae. The range varies from user to user, but in general, they are capable of seeing from a minimum of 10 meters, all the way up to a maximum of 10 kilometers, and the amount grows with training, ability and age.
This vision is telescopic, allowing users to see through solid objects such as Smoke screens and walls. Arguably its greatest ability is to see chakra in all its nature, including its flow and the circulation system inside the body with great detail as well as the colour of chakra; giving them the ability to discern clones from the original. Like the Sharingan and Rinnegan, the Byakugan can detect chakra sealed barriers as well as determine when genjutsu is being used.
It is from the Byakugan's especially acute perception of chakra and ability to view the tenketsu of the human body that the Hyūga Clan are able to employ their signature fighting style: Gentle Fist. Able to strike without force, and yet do more damage to a ninja than some of the most powerful ninjutsu, the Gentle Fist has been labelled as the greatest taijutsu style known in all of Konohagakure.
The most famed style of Konohagakure no Sato and undoubtedly the style in which relies on light strikes, hence the name; but incredibly powerful at dispatching foes. It is a close hand to hand combat that works on strikes, blocks and redirection. The style works by attacking an opponent's chakra pathway system, which are closely interlinked with the surrounding organs and muscles. Even the slightest tap can cause severe damage to the organs, giving it's namesake.
This is done by the user forcibly injecting their own chakra into the opponent's system, damaging the surrounding organs/muscles/etc.,This is usually performed through the index and middle finger or through their palm. With the aid of the Byakugan they can gain an intricate understanding of the tenketsu, and can open, or close the Tenketsu. This wrecks havoc in itself as inflicting these injuries completely changes the dynamic in people's chakra systems that allow them to mould chakra, that can range from worse chakra control to zero ability to perform ninjutsu.
It should be noted, that any chakra based substance can be destroyed by the Gentle fist, as done by leaking chakra through into their hands into the shape of a protruding needle, or knife-like blade to cut through the opposing chakra.
Because the chakra pathway system is invisible to the naked eye, the Byakugan is required for this style to be used effectively. Since the Byakugan is unique to the Hyūga clan, it has become their signature style of combat.