Just as the fact that a categorification of $\mathbb{N}$ is the category of finite dimensional vector spaces, a categorification of $\mathbb{Z}$ (in my mind) is the category of bounded cochain complexes of finite dimensional vector spaces which takes the Euler characteristic as the decategorification map.

Is there any paper which talk about more details on this construction?

Baez-Crans $2$-vector spaces(math.ucr.edu/home/baez/hda6.pdf), which as a search term should give more things to look at. – Qiaochu Yuan Sep 11 '12 at 5:52youshould write a paper on it! ;-) You could also take the easier category of $\mathbb{Z}/2$-graded vector spaces. – Fernando Muro Sep 11 '12 at 6:58