You generally do an axiomatic definition:
A field F is said to be ordered if it satisfies the following axiom:
There is a nonempty subset P of the field F (called the positive subset) for which
(i) If a and b are in P, then a + b is in P (addition closure)
(ii) If a and b are in P, then ab is in P (multiplicative closure)
(iii) For any a in F exactly one of the following holds: a is in P, -a is in P, or a = 0 (trichotomy).
Then you define, for ordered field F and its positive subset P, the < relation as follows:
Let a, b be in F. We say that a < b if b - a is in P. a < b and b > a are equivalent.
So it's a bit axiomatic, usually. There might be an even more rigorous definition in your set theory books. I've given you the definition on pages 16-7 of Kirkwood's An Introduction to Analysis.