Q:

How can flux be the number of electric field lines normal to a surface?

Shouldn't the number of field lines through any surface be infinite if there is a charge present. Or is there a finite number of field lines from a unit charge?

A:

What you have to understand is that field lines have no physical significance and are merely something we use to conveniently represent Electric Fields. It doesn't matter whether a million field lines or only one passes through your surface. They represents the exact same thing; as long as your field and configuration remain the same.
The flux of any vector field through a surface is the dot product of the Field and Area vectors. The definition which you state is not correct. 
You should check out the answers to this question, they're great: Why does the density of electric field lines make sense, if there is a field line through every point?

A:

 
I hope this helps you. Also the electric field lines of force are purely a geometrical construction, which helps us to visualize the nature of electric field in a region.

