MILP problem NP-hard proof
In general, a general MILP was proved to be NP-Hard. This is a well-known proof and can be easily found in any textbook (or even on Wikipedia).
However, you can see one specific MILP problem as an 'instance' of the general MILP problem. Thus, you still need to show its complexity.



In fact, several MILP problems are NP-Hard:


Elementary Shortest Path (if the graph has negative cycles)
Degree-constrained Minimum Spanning Tree
Vehicle Routing Problem


However, many others are shown to be solvable in polynomial-time:


Job scheduling
Minimum Spanning Tree
Aircraft scheduling problem




You can demonstrate that a MILP problem is NP-Hard in the same way you do with any other optimization problem. 

All you need to do is to reduce an NP-Hard optimization problem $\sigma$ to your MILP, construct a solution for MILP, and shows a polynomial-time transformation of your obtained solution to a solution of $\sigma$.  

Sadly, there is no standard technique or proof method that can be applied to show the complexity of any MILP problems and you need to use your creativity (and mathematical skills, in many cases) to proof the complexity of your problem.