Why current flows in an inductor? If during self-induction the inductor produces a current in opposite direction of that of battery in order to resist the change of current through it then how could the current after some reasonable amount of time gets to its peak value? I mean there should be no net current flowing through the circuit since at starting the net current was 0 and the property of inductor is to resist the change in electric current.
 A: The whole point is that for a normal conductor Lenz can never win.  
It is true that at the start there is no current but there is a finite rate of change of current with time and so with the passage of time the current will change from its initial zero value.  
Another way of looking at the situation is that if Lenz did stop the change of current, and hence magnetic flux produced by the current,then Faraday would not produce an opposing emf as there had been no change of flux and that does not happen because then the applied emf will be able to change the current..
A: 
If during self-induction the inductor produces a current in opposite direction of that of battery in order to resist the change of current through it then how could the current after some reasonable amount of time gets to its peek point?

If you apply a DC voltage to an ideal inductor, it will never reach its peak current. The current will continue to increase indefinitely. 
Only if you consider real voltage sources (which have internal resistance) or real inductors (which have non-zero resistance) will there be a limit on the current.

I mean there should be no net current flowing through the circuit since at starting the net current was 0 and the property of inductor is to resist the change in electric current.

Resistors also resist the flow of current by producing a voltage opposing the current. And yet current can flow through resistors. 
The answer is that "opposing" or "resisting" does not mean "completely stopping". 
In the case of the resistor, it means that the current developed when a voltage is applied is finite (and determined by Ohm's law) rather than infinite.
In the case of an inductor, it means that the rate of change of current when a voltage is applied is finite rather than infinite. It doesn't mean that no current can flow at all.
