OK, here we go ...
First, he isn't a natural-born citizen at all, he is a native-born citizen.
There is a difference between a 'citizen' and a 'natural born citizen'.
A 'citizen' can be a naturalized US citizen by immigrating to the US and applying for citizenship, a native-born citizen by just being born in the US, or a citizen by descent by being born to a US citizen parent.
A "natural-born citizen" is someone who was born to two US citizen parents on US soil.
Rep. John Bingham of Ohio, chief architect of the 14th Amendment's first section, considered the proposed national law on citizenship as “simply declaratory of what is written in the Constitution, that every human being born within the jurisdiction of the United States of parents not owing allegiance to any foreign sovereignty is, in the language of your Constitution itself, a natural born citizen…” (congressional globe, 39th congress, 1st session, March 9, 1866)
"Not owing allegiance to any foreign sovereignty" is the key phrase here. Both parents had to be US citizens, neither could be non-US citizen because then they would owe allegiance to a foreign sovereignty.
The Supreme Court case of PERKINS v. ELG (1939) defined a natural born citizen as a person who was born on US soil to two US citizen parents.It defined it in two places; once by declaring Ms. Elg a "natural born citizen" (she was born to two Swedish immigrant naturalized US citizen parents on US soil) and once by citing the Steinkauler case, in which a young man was born on US soil to two naturalized US citizen parents and said of him "in due time, if the people elect, he can become President of the United States"
See PERKINS V. ELG - http://caselaw.lp.findlaw.com/scripts/getcase.pl?court=US&vol=307&invol=325
It was also defined in MINOR v. HAPPERSETT (1874) http://caselaw.lp.findlaw.com/scripts/getcase.pl?court=US&vol=88&invol=162 where the court said "At common-law, with the nomenclature of which the framers of the Constitution were familiar, it was never doubted that all children born in a country of parents who were its citizens became themselves, upon their birth, citizens also. These were natives, or natural-born citizens ..."
In the Supreme Court case of Kwock Jan Fat v. White, 253 U.S. 454 (1920) The father of Kwock Jan Fat was a native-born US citizen, born on US soil of Chinese nationals. The mother of Kwock Jan Fat was a naturalized US citizen by virtue of her marriage to a US citizen. http://supreme.justia.com/us/253/454/case.html
Kwock Jan Fat was Born of two US citizens on US soil and thus a "natural born citizen."
To be eligible to serve as the President of the United States, the person must be a 'natural born citizen' (or a citizen at the time of the adoption of the constitution) per Article 2 Section 1 of the US Constitution. http://www.law.cornell.edu/constitution/constitution.articleii.html
Barack Obama's father was not an immigrant to the US. He was in the US on a student visa and was a British/Kenyan citizen. He never had "legal resident" status and he never applied for US citizenship.
It is not Unconstitutional for an ineligible person to be a candidate for President, or even win an election. It is Unconstitutional for an ineligible person to "serve" as President of the United States. Since he has been inaugurated and is now serving as President, a case may be heard before the Supreme Court on the issue of his eligibility.