When was Abraham alive? It is generally accepted that Abraham lived 175 years. However, the dates when he lived are rather fuzzy and hotly debated. 

When was he alive and what reasons would you give for your conclusion?

What methods have gone into determining the dates in which Abraham (and perhaps other early biblical figures) lived and died?
The best way to find this estimate is to take known historical dates and work backwards from there using dates and durations in the Bible.

The Mesha Stele has been dated to about 840-850 BCE, and seems to pretty clearly describe the time of Omri.  If we assume it was written when the events happened, not long afterwards, we can use it to work backwards.  Omri's reign began in the 31st year of Asa and lasted twelve years, which means that Asa's reign began around 870-890.  Previously, Rehoboam ruled for 17 and Abijah for three.  Although the transition is not entirely clear to me, the prior ruler was apparently Solomon for 40 years.  In his fourth year, he started to build the temple, which was 480 years after leaving Egypt.  So that takes us back 20+36+480 = 536 years further, to 1406-1426 BCE as the dates for the exodus.

Now we come to a difficult stretch because there are, to my knowledge, no clear chronological links between Moses and, say, Joseph.  Moses' grandfather Kehat was with Jacob on the way into Egypt, and if you add up all the ages that's a maximum of 350 years (actually surely less, since Kehat was not an infant), but it's not clear what the minimum was.  The most direct statement is Genesis 15:13, which puts four hundred years between Abraham's offspring and the exodus.  Since the phrase is "in a land that is not theirs", and Ishmael and Isaac did not start off owning the land, it seems that the most harmonious interpretation is that this was 400 years from Ishmael's birth.  We find that Abraham was 86 years old at the time.  The 400 years probably didn't end until after wandering in the wilderness for 40 years, so that's 360 years from Ishmael's birth to the exodus.

Therefore, according to historical and Biblical evidence, Abraham was born somewhere in the range of 1852-1872 BCE and died 175 years later (1677-1697).  If the 400 years are accounted for differently, these dates could change significantly.

For what it's worth, other people have studied the matter and come up with different timelines; it's complicated by the lack of accurately-dated historical evidence for events prior to David upon which to anchor the timeline.

(Beware of lots of little edits to fix up small mathematical errors and other details.)