I want to know whether the principal congruence subgroups of $SL(n, \mathbb{Z})$ are characteristic? please suggest me a reference.
Yes, they are caracteristic. Let $\rho: GL(n, \mathbb{Z}) \to GL(n, \mathbb{Z}/m\mathbb{Z})$ be reduction mod $m$ and $\Gamma_n(m) := ker(\rho) \cap SL(n, \mathbb{Z})$ be a congruence sugroup. According to the discussion in Automorphisms of $SL_n(\mathbb{Z})$ the automorphisms of $SL(n, \mathbb{Z})$ for $n > 2$ are generated by
Obviously, $\Gamma_n(m)$ is invariant under these automorphisms and thus is characteristic. In case $n = 2$ there is one more automorphisms of $SL(2, \mathbb{Z})$ that also leaves $\Gamma_n(m)$ invariant (for a description of this automorphism see the paper of Hua and Reiner mentioned in the link above). 


I think the answer is yes. See this question for a discussion of $\mathrm{Aut}(\mathrm{SL}(n,\mathbb{Z}))$. The generators given in the HuaReiner paper seem to preserve the principal congruence subgroups. 

