After Book 6, when Snape killed Dumbledore, as per the magical law every Hogwarts headmaster/headmistress gets his/her portrait in the headmaster's office. This was known to Harry and obviously he was aware that Dumbledore will have his portrait in the headmaster's office as well.

We also know that portraits of the headmaster can travel between their portraits outside the headmaster's office as well. For instance, Phineas could travel between Hogwarts and Grimmauld Palace. Similarly, Dilys Derwent can travel between Hogwarts and St. Mungo.

We also know that headmasters in the portraits can travel between the portraits irrespective of the actual location of the portrait. For instance, when Hermione took out Phineas portrait from Grimmauld Palace and put it in her purse, she was able to interact with it when needed. Moreover, Phineas was also able to listen to their conversation whenever the purse was opened.

Now this seems very far fetched that Dumbledore didn't discuss about this with Harry when he was giving him lessons in Book 6. Dumbledore knew perfectly well that he had only few months left and Harry will definitely need his help and guidance when he will be hunting the horcruxes.

Why didn't he advise Harry to keep a portrait of him along with him in case something happens to him?

This seems like a convenient flaw in the plot to make the horcrux hunting a lot more laborious work than it actually was.

Furthermore, why Harry couldn't think of this when he started hunting the horcruxes in Book 7?

He knew perfectly well that he had absolutely no idea about the location of the horcruxes let alone what to do with them and how to destroy them when he found them. That was the reason the trio kept on dangling the locket in their necks for months long. It could have been a lot easier and much better approach to get the portrait of Albus Dumbledore along with them and getting advise from it wherever needed.

marked as duplicate by Valorum harry-potter Oct 26 '17 at 10:05

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    The question of what Dumbledore could have added to Harry's search had he been able to communicate with him (spoiler, not a lot) is covered in both dupes. – Valorum Oct 26 '17 at 10:06
  • And another problem with your idea: how exactly do you think Harry would have been given the sword if he had the portrait? Dumbledore overhears Phineas talking about where they are; Dumbledore then instructs Severus to get the sword to him. Remember Severus is also in the dark about the Horcruxes. And maybe it's because Harry didn't feel it was his right to take the portrait or didn't think about it (He was sort of in shock and grieving!). And as for in book 7? He had no choice at that point did he? You also must remember they're still kids (even if they come of age they're still quite young). – Pryftan Oct 26 '17 at 18:03
  • @Pryftan - I don't think you got the question properly. I would suggest you to read it one more time, perhaps you didn't get the query earlier. And AFA sword is concerned, if you remember, Dumbledore portrait WILL ALWAYS be in the headmaster's office also. So Dumbledore can move between those portraits and instruct Snape freely where needed. My point is, the trio can talk and get help from many long gone individuals from their portraits on regular basis. This seems like a flaw since the only people he can't get help is his family and friends. – Harry_S Oct 27 '17 at 9:05
  • @Harry_S Exactly what other portrait of Dumbledore are you referring to then? Or am I really forgetting it that much? The reason Phineas can move between is that there IS another portrait of him: In Grimauld's Place. The only portrait of Dumbledore is in Hogwarts and thus if the trio had it it'd not be in Hogwarts. Or are you saying by moving it it's duplicated? I don't recall seeing that. Incidentally your argument doesn't cover the remainder of my points. – Pryftan Oct 27 '17 at 15:57