This problem can be solved easily by using Table. Please see the documentation on Table for examples of how to use it.

pointList = Table[

A = x^3 + 3;

B = 3 x;

Equation = z^2 + A*z + B;

V1 = Re[Root[Equation, 1]];

{x, V1}, {x, 0.1, 10, 0.1}]

You can then use ListPlot on the result:

ListPlot[pointList]