Dumbledore set up his own death as a part of big plan. He died successfully. Why didn't he come back as ghost "secretly"? His ghost couldn't use spells, but he would still likely have retained encyclopedia of vast knowledge and wisdom. If Professor Binns's memories can remain intact, I don't see any reason why Dumbledore's ghost couldn't assist Harry and good guys in secret.
Harry answers this in Deathly Hallows:
‘I can’t – but can you explain how Dumbledore sent him to us if he’s lying in a tomb at Hogwarts?’
‘I dunno, it could’ve been his ghost!’
‘Dumbledore wouldn’t come back as a ghost,’ said Harry. There was little about Dumbledore he was sure of, now, but he knew that much. ‘He would have gone on.’
‘What d’you mean, “gone on”?’ asked Ron, but before Harry could say any more, a voice behind them said, ‘’Arry?’
This comment seems to be an echo of Harry's conversation with Sir Nicholas in HP:OotP chapter 38. In short, Harry considers it vanishingly unlikely that anyone as vital as Dumbledore would choose to become a "pale imitation of life"
Wizards can leave an imprint of themselves upon the earth, to walk palely where their living selves once trod,’ said Nick miserably. ‘But very few wizards choose that path.’
‘He will not come back,’ repeated Nick. ‘He will have … gone on.’
‘What d’you mean, “gone on”?’ said Harry quickly. ‘Gone on where? Listen – what happens when you die, anyway? Where do you go? Why doesn’t everyone come back? Why isn’t this place full of ghosts? Why –?’
‘I cannot answer,’ said Nick.
‘You’re dead, aren’t you?’ said Harry exasperatedly. ‘Who can answer better than you?’
‘I was afraid of death,’ said Nick softly. ‘I chose to remain behind. I sometimes wonder whether I oughtn’t to have … well, that is neither here nor there … in fact, I am neither here nor there …’ He gave a small sad chuckle. ‘I know nothing of the secrets of death, Harry, for I chose my feeble imitation of life instead. I believe learned wizards study the matter in the Department of Mysteries –’
As we all know, in Deathly Hallows, Harry walked towards death. Voldemort shot him with Avada Kedavra so he fell to the ground. He went to, let's say, heaven and saw Albus Dumbledore as a ghost. Dumbledore asked him where he thought he was, and he said King's Cross Station. Dumbledore later explains that dead people come here and if they want to stay dead, not as a ghost, they could jump on the train. It is later revealed that Harry was not dead but unconscious.
Dumbledore was a brave man. He would never have chosen the path for the weak and desperate.
James and Lily Potter chose not to remain behind because they knew they had friends and other relatives (Petunia) that would take care of Harry. They also knew that had they come back, they would regret it in later life.
Sirius Black knew he would die and he was ready for it. After all, he is a wanted yet innocent criminal (though no one knew he was innocent).
Severus Snape chose to take care of Harry for Harry's mother's sake. He loved her but because of his resemblance to James, he hated Harry. As we know, Harry named his son Albus Severus after Snape because he is the "bravest man he ever knew".
Remus Lupin knew the world would be better off without him.