How did the Cullens maintain their identity as human beings in the real world?
Schools and colleges require past records and birth certificates, and they leave a trail of records too. As a doctor, Carlisle would need an updated degree every time he had to start over because his age and appearance did not make sense. And, doctors build renown. How did the Cullens manage all this?
How vampires used to manage to maintain their identity as human?
They have their illegal legal paper and ID dealings with J.Jenks
They don't always move to big cities together. They are careful and move separately or in small groups.
Carlisle doesn't practice in big cities, he has mostly practised as a Doctor around small towns.
Whose to say that they don't get J. Jenks to clear all the evidence about them in certain cities?
They let things settle down before moving back. They waited a while before moving back to Forks, to let people die and forget them. They do this everywhere they have went.
It Appears from the Twilight Wikia: Jasper's page that Jasper maintains the relationship with J.Jenks the man that forges them ID's, and legal papers that helps them survive in the "human world" and appear very legal.
J. Jenks, better known as J, is a lawyer and forger of illegal documents, and Jasper is his foremost client. Jasper knew his former partner fifteen years before they met. Whenever his family needs new legal paperwork, he would come to J. for forgery.
J. is very intimidated by Jasper due to his limited knowledge of Jasper's immortality. In Breaking Dawn, Alice gives Bella clues to lead her to J. to help her forge passports and ID's for Renesmee in case they need to escape the Volturi. Afterward, Bella decides to take over Jasper's business relationship to J. to ease his stress.
They aren't above the law.
Carlisle is a doctor and with all the help he does he would probably be found out by now, but he goes to small towns.
Much later, Carlisle and his family moved to Forks, Washington where they had lived before (shortly after Emmett and before Alice and Jasper joined) and had made a treaty with the Quileute tribe, led by a shape-shifter named Ephraim Black. The treaty stated that the Quileutes would leave the Cullens alone and not reveal them to humans if the Cullens would refrain from biting humans - whether to kill or change them - or trespassing on Quileute land. Because of the treaty, the Cullens were able to live in relative peace.
The whole family even goes on trips(not together) to hide from publicity that would see their true nature.
From the Twilight Wikia about Emmett and his wife:
They rarely separated after this; the two having found in the other their true soul mate. Edward had once told Bella that they were always together, and it was hard to be in a five mile radius of them. The pair would marry often and sometimes lived separate from the rest of their adopted family so as to live as a newlywed couple. In 1937, while living in Forks, they were discovered by Ephraim Black's pack of shape-shifters. To secure both sides, the Cullens and the pack made a treaty: they shall not attack or expose the Cullens unless they bite a human or cross over to the Quileute traditional lands. They moved away sometime after the agreement was made.**
According to the Twilight Wikia, we learn in the Breaking Dawn, Part II that Edward Cullen has employed a law firm since the late 1960s, in part to help him procure fake ID documents and other paperwork needed to ensure that no-one detects his age. It's reasonable to assume that other vampires have similar arrangements.
Jenks : I've been working with them for more than 20 years. And my late partner knew Jasper 15 years before that. He is, uh, unusually well-preserved.
The lawyer he meets, J.Jenks and his partner before him are clearly aware that there's something unusual about this (ageless) client but it's not immediately clear whether they're aware that he's a vampire or if they just accept at face value that he simply looks unusually young for a 50 year old.
As to how to avoid building a reputation, my guess would be that vampires periodically change professions in order to "go to ground". After a few decades, those who were peers (and would have any first-hand knowledge of them) would have either died, left their jobs or simply wouldn't expect to see someone they knew 40 years ago to look the same. Moving physical location, especially changing countries, would also be a major way of removing yourself from a peer group.
