Ho claims that Vietnam's independence is consistent with the philosophical principles which the Allies claimed were paramount during World War II. What principles was Ho referring to, and does he make references to occasions where those principles were reasserted?

Ho Chi Minh is highlighting specific principles that were established in the Atlantic Charter of August 14th, 1941.  The Atlantic Charter was a post-war plan that was to be enacted at the conclusion of World War II.  Some of the more pertinent points in the charter are as follows: 

“First, their countries seek no aggrandizement, territorial or other;
Second, they desire to see no territorial changes that do not accord with the freely expressed wishes of the peoples concerned;
Third, they respect the right of all peoples to choose the form of government under which they will live; and they wish to see sovereign rights and self government restored to those who have been forcibly deprived of them
Fourth, they will endeavor, with due respect for their existing obligations, to further the enjoyment by all States, great or small, victor or vanquished, of access, on equal terms, to the trade and to the raw materials of the world which are needed for their economic prosperity;
Fifth, they desire to bring about the fullest collaboration between all nations in the economic field with the object of securing, for all, improved labor standards, economic advancement and social security.” (Source)

By analyzing these statements, one can see how Ho Chi Minh can apply them to the notion of independence from France.
France during World War II was completely overrun and controlled by Germany in what appears to be a similar fashion to how the French colonized Vietnam through economic imperialism.  The French were getting a taste of what the Vietnamese had felt under the yoke and rule of the French.
Ho Chi Minh reasserts these ideas in the Vietnamese Declaration of Independence by referencing not only the American Declaration of Independence but also the French Declaration of the Rights of Man and of the Citizen.

"All men are created equal. They are endowed by their Creator with certain inalienable rights, among these are Life, Liberty, and the pursuit of Happiness" This immortal statement was made in the Declaration of Independence of the United States of America m 1776. In a broader sense, this means: All the peoples on the earth are equal from birth, all the peoples have a right to live, to be happy and free. The Declaration of the French Revolution made in 1791 on the Rights of Man and the Citizen also states: "All men are born free and with equal rights, and must always remain free and have equal rights." Those are undeniable truths. Nevertheless, for more than eighty years, the French imperialists, abusing the standard of Liberty, Equality, and Fraternity, have violated our Fatherland and oppressed our fellow-citizens. They have acted contrary to the ideals of humanity and justice. In the field of politics, they have deprived our people of every democratic liberty.”

Source - New China News Agency, April 6, 1951.
As you can see, it was an effective tactic to make a profound point using the very words of their oppressors against them.
Ho Chi Minh highlights the ideas that all men are created equal and that freedom must be guaranteed to the people.  By pointing out that the French imperialists have abused the ideals of liberty and equality, he justifies independence by the same means the French did during the French Revolution as well as the Americans during the American Revolution.  The French would have a hard time justifying their economic control over Vietnam after what they recently experienced with Hitler's brutal takeover of France.
So by examining the declarations of independence as well as the Atlantic Charter, you can see where Ho Chi Minh's claims are not only appealing but logical as well.
https://avalon.law.yale.edu/wwii/atlantic.asp