Walter tells Victoria that he was brainwashed and forcefully turned into Millenium's minion, but then says that he attacks what remains of Hellsing by his own free will. Bad characters usually mess with good ones, so it's not clear how it really happened to him.
From the Hellsing Wiki:
It is also currently unknown whether or not Walter's brainwashing by Millennium was consensual, or if he was brainwashed against his will.
However, on the same page, it is also stated that
In chapter 75 of the Hellsing manga, Sir Hugh Irons, the acting leader of the Convention of Twelve, comes to the conclusion that Walter had in fact been betraying his superiors for quite some time, even before the second coming of Millennium and possibly even since the days of the World War II. He mentions that he foresaw Richard Hellsing's attempted coup of the Hellsing Organization and also Richard's assassination attempt on Integra, and had quite specifically warned Walter of it, charging him with protecting her; and yet, despite his warning, Walter was nowhere to be found on that day. Therefore it is suggested that Alucard's awakening was also orchestrated by Walter; his convenient absence left Integra with no other choice but to release Alucard, whom Arthur Hellsing (Integra's father) had purposefully sealed away, thinking him to be too dangerous to use as a weapon. Other bits of evidence suggesting that Walter is a traitor include the Valentine brothers' mysterious knowledge of the location of the Hellsing mansion, Walter's suspicious memory lapse when Millennium resurfaces (he does not seem to remember storming and destroying Millennium's base in World War II, but claims that he does after Alucard prods him), and the strange omission of the fight scene between Walter and his apparent rival, the Captain. It may be that, after seeing Alucard's abilities during World War II, Walter wished to destroy him to prove himself and thus allowed Alucard to awaken.
It's still shrouded in mystery, however, I think he wanted to battle against Alucard in a fair fight, more than "betray" Integra or Hellsing.
He may have been brainwashed and he may have not, but it was very clear that he wanted to beat Alucard in battle, and that the deal he made with Millennium was made to achieve that goal.
Actually, Walter wasn't brainwashed and he wasn't exactly Millennium's servant either, if not he would not have protected Integra. He only wanted to do his job as a vampire hunter, and kill Alucard. He first met Alucard during World War II; they were allies then. He knew he wasn't strong enough at that time to match strength against Alucard, since he was just a teenager. So, he made a deal with the Major in 1944, probably during Hellsing: The Dawn. A deal that he would be turned into a vampire, so that he will be strong enough to match Alucard.
However, Alucard was sealed away after World War II by Integra's father. It took 50 years, but Walter made sure that Integra had no choice but to release Alucard upon her father's death. Walter knew that Integra's uncle would try to kill her, and she would have to release Alucard if he wasn't there to defend her. But that was still not enough, he knew that Alucard had millions of lives within him, and he couldn't be killed, so he had to wait until Alucard was more vulnerable. He waited 5 years, that's when Millennium invaded London, and Alucard was given the order to release Level 0, releasing all the lives within him, leaving him with 1 life left within him, that when Alucard was at his most vulnerable.
But all that was for nothing, Alucard just summoned all the blood on the battlefield to him during the fight, reclaiming all the souls that he had released at Level 0, plus the souls of those who had died on the battlefield that day, giving him about 3.5 million lives, which was more lives than at the beginning.
The answer to this question is so much more simple than you guys are making it. Yes, he says to Seras after she asked what happened to him that he "was brainwashed and forcefully turned into Millenium's minion," but then immediately after goes on to say "Is that what you wanted to hear?" implying that everything he said before that was complete and total sarcasm. He did it willingly. No one knows the true reason he betrayed the Hellsing Organisation. However, Alucard does say in his fight with Walter that he thought it was because Walter was "afraid of being old and useless. Maybe afraid of being forgotten." Walter confirmed Alucard's theory of being forgotten by saying
We are the evening's entertainment. And I... just wanted to do something with my time on the stage worthy of applause...
Which can lead us to believe that Walter wanted to do something he would be remembered by. And killing Alucard would certainly be remembered. But he never specifically says that's why he betrayed them
So why Walter betrayed the Hellsing Organisation is unknown, but Walter very clearly says it was his own choice. When he said he was brainwashed, he followed it up with something implying that that comment was to be taken as sarcasm.