Question

what claim does Jefferson make in the first paragraph of the Declaration of Independence, and how does he say he will support that claim?

Answer 1

The Declaration of Independence, in general, justifies the independence of British colonies. The first paragraph states it and confirms that why the separation from British Empire was that important. The paragraph claims that there are certain reasons for that. And in the consequent paragraphs, Jefferson explains those reasons. The document states that all men are created equal and therefore not any institution, even a government cannot violate their rights.

In the second paragraph, Jefferson counts those rights. They are rights to Life, Liberty and the pursuit of Happiness. Even in the document itself, those words have been written with capital letters to emphasize their significance. Therefore, all the rights have to be respected and when it is not respected, the situation cannot continue further. In this case the older government should be replaced with the new one. It is a new government which can respect and protect those rights.

Answer 2

As history changes, and people/nations evolve, it becomes necessary for them to be independent of any other country/power on which they relied. They feel the need to break those connections that bound them, and to finally emerge as a country of its own, a country which doesn't need to count on others in order to thrive and prosper. God has given everyone equal rights, and thus no country has more rights than any other. This newly founded country will become respectable through its own effort, and not through the help/or the dominion of another country. The Americans felt impelled to separate from their mother country, Britain, and eventually create a dream of their own. 
All men are equal - God has given them those unalienable rights. The people are those who make decisions - there is a Government, but if it becomes destructive, it is the people who decide how to organize the country. 
Jefferson talks about the downsides of being a British colony, and thus creates a chance for the Americans to create a country in which these rules needn't be followed so blindly. They are free to do what they wish, because it is their God-given right.