When and how was carbon first formed on Earth?

Carbon was not formed on earth.

Some of the carbon on earth is probably about 15 billion years old and dates back to the Big Bang. Some of it may have been created much later, as recently as 4.5 to 5 billion years ago when the Earth was formed.

http://pubs.usgs.gov/gip/geotime/age.html

http://www.umich.edu/~gs265/bigbang.htm

It was formed in a sun or star by a nuclear fusion reaction. At some point the sun or star exploded and the material that it had created became unconsolidated asteroids and space dust particles that gradually became attracted to each other by gravity, eventually forming a planet.

http://www.cfa.harvard.edu/COMPLETE/learn/planets/...

How Is Carbon Formed

Formation of the carbon atomic nucleus requires a nearly simultaneous triple collision of alpha particles (helium nuclei) within the core of a giant or supergiant star. This happens in conditions of > 100 megakelvin temperature and helium concentration that the rapid expansion and cooling of the early universe prohibited, and therefore no significant carbon was created during the Big Bang. Instead, the interiors of stars in the horizontal branch transform three helium nuclei into carbon by means of this triple-alpha process. In order to be available for formation of life as we know it, this carbon must then later be scattered into space as dust, in supernova explosions, as part of the material which later forms second, third-generation star systems which have planets accreted from such dust. The Solar System is one such third-generation star system.

Carbon was present at the formation of our solar system and therefore our planet as well