Dragon priests are liches are not technically the same. Dragon priests are granted eternal life by the dragons (I think, either that or the gods damned them to never be able to die) and liches are necromancers who used a form of necromancy to prolong their life.

Do liches and vampires get along with eachother?

I would say Vampire Lords, powerful Liches and Dragon priest has stronger magic but vampire lords could use more tactics like sneak, melee powers, and speed.

So there techniquely the same except that they achieved there lichdom in different ways?

Liches are necromancers which have embraced the power of Lichdom, placing their soul in an object called a 'Phylactery', which is usually a Colossal Black Soul Gem. They are always selfish and power-hungry, destroying all in their searches for souls to repair the Phylactery. They are extremely intelligent and are adept spellcasters; along with their random effect staffs, they carry tremendous power and arcane magic. Nether Liches were normal liches, but, for a certain time were weakened by a destruction of one of their Phylacteries. Ancient Liches created more then one phylactery, becoming much more powerful. A Lich is immortal and powerful, always resurrecting if their Phylactery is not crushed. They always have control of at least a small undead army and a tomb.

Do liches and vampires get along with eachother?

Nope, the only Lich i know with vampire allies is the King of Worm.

I don't understand why they hate eachother i mean both are immortal are the liches just jealous that vamps get to keep there good looks and don't smell bad?

Well, since Liches and Dragon Priests are the one and the same, it would be more of a Vampire Lord vs a Lich, Dragon aligned or not. We simply don't know enough about the Lords to compare.

I don't understand why they hate eachother i mean both are immortal are the liches just jealous that vamps get to keep there good looks and don't smell bad?