How the Trail of Tears violated the pursuit of happiness?

Answer 1

The actions taken by the Indian Removal Act violated several provisions of the US Constitution, these violations are relational so may take a bit to explain.

The First or biggest violation was under article 6 of the Constitution which states, in relevant part "This Constitution and the Treaties made by the authority of the United States, is the Supreme Law of the Land and [list of all state & federal members of government] are bound by oath to support and defend it.

Since the Native Americans were defined as the legal owners, and independent governments of / on the lands of North America (See: De India et De Jure Belli Reflections, Francisco de Vitoria cir. 1532) and governments were required to form Treaties (contracts of sale) for the land - the United States violated prior treaties that promised the Cherokee (among others) that their first removals were final and would not happen again.

The United States attempted to avoid this problem by re-defining the Native Americans as domestically dependent governments (states) in "Federal Territories" - this implies constitutional protections of Property rights and it is a violation of the constitution to deprive a person of property without just compensation (even the first treaties failed to provide compensation) - and the false treaties, granting lands in what is today modern day Oklahoma, were violated. It should also be noted that international law (Vitoria) enforced these requirements, and the USA ignored these laws.

Further, the United States violated the aforementioned treaties which is possible ONLY if you have the agreement of the other party (which they did not), and this is known as Unjust profiteering (fraud) and requires (under law) disgorgement (a return of property taken). The US Court system has refused any attempt at a suit under these causes, including any suit before the International Court of Justice (another Treaty Violation; multiple UN treaties).

Answer 2

Probably not insofar as government action malfeasant or not, is rarely consideed to be a violation of this right.

If I were to seek Constitutional grounds against the Trail of Tears action, I'd probably look more at 4th Amendment.