I provided a direct axiomatization of integers for instance under the MO question Axiomatic definition..., in the post Part 2: Cyclands and integers. This axiomatization makes no direct reference to the natural numbers, or to any linear order.
On the other hand, a clean direct constructions of integers can be done as follows:
each integer is uniquely defined as a sequence $(a_0\ a_1\ \ldots)$ of integers (called digits): $-1\ \,0\ \,1$, where all digits but a finite number are equal to $0$.
(The rest is obvious). By using only the first $n$ digits we get the continuous range of $3^n$ integers: $$-\frac{3^n-1}2\ \ldots\ 0\ \ldots\ \frac{3^n-1}2$$
Furthermore, the ring $\mathbb Z[\frac 13]$ can be defined as all sequences $f:\mathbb Z\rightarrow \{-1\ \,0\ \,1\}$ such that all values $f(n)$ but a finite number are $0$.