Problems understanding membrane potential
I think this question has more to do with kinetics / transport phenomenons than biology, but that's okay, everything is connected especially my computer to the internet. ;-)

The basic idea behind transport phenomenons is that there will always be a flux of quantitative properties (e.g. charges, particle number, entropy, volume, etc...) where the qualitative properties like (electrical potential, chemical potential, temperature, pressure, etc...) have a gradient in space (note that kinetics of chemical reactions can be described similarly without the space gradient part).

In this case we are talking about electrochemical potential gradient and ion flux. It is very important to recognize that the electrochemical potential is not the same as the electric potential. It has a chemical component, so the result will depend on both charge and concentration gradients. By a single ion channel system (e.g. Na⁺ channel only) you can count the potential of each side of the membrane using the Nernst equation, by a system with multiple ion channels it is more complicated, so in that case you have to use the Goldman equation. So at the end you can say something like one side has a potential of x[mV] and the other side has a potential of y[mV] and so the difference is:  d[mV] = x[mV] - y[mV] 

The sign of d depends on x or y has a greater value. By single ion channel systems this expresses an x → y direction so by positive d the cation flow direction is x → y and by negative d it is y → x. By anion flow it is the opposite.

By cells we use the out → in direction by counting the membrane potential. We can count the potentials using the 


Nernst equation:
$$E_m = \frac{RT}{zF} \times ln \left(\frac{c_{out}}{c_{in}}\right)$$ or the 
Goldman equation: 
$$E_m = \frac{RT}{F} \times ln \left(\frac{p_{K^+}.[K^+]_{out} + p_{Na^+}.[Na^+]_{out} + p_{Cl^-}.[Cl^-]_{in}} {p_{K^+}.[K^+]_{in} + p_{Na^+}.[Na^+]_{in} + p_{Cl^-}.[Cl^-]_{out}}\right)$$
where  


$Em$ is the membrane potential
$z$ is the ion charge
$[X]$ is the ion concentration
$p_X$ is the relative membrane permeability for the actual ion


and so on...


So for example by an average human cell in blood plasma, assuming that the inside of the membrane has a small negative charge


the Na⁺ has a positive effect (positive charge, high out → in concentration gradient, low out → in charge gradient),
the K⁺ has a negative effect (positive charge, high in → out concentration gradient, low out → in charge gradient),
the Cl¯ has a negative effect (negative charge, high out → in concentration gradient, low in → out charge gradient)
the Mg⁺⁺ has a small positive effect (positive charge, low out → in concentration gradient, low out → in charge gradient)


on the membrane potential. It is hard to find an ion in this case which flow is charge gradient dominated instead of concentration gradient...


  I understand that membrane potential is the difference of the
  extracellular and intracellular ionic charges, due to their
  concentrations.


The (sign and) value of the membrane potential is not determined by the charge gradient because the concentration gradients usually have a much bigger effect on it. The cells maintain these concentration gradients by using up energy.


  We work with a regular cell with potassium 120mM inside and 4.5mM
  outside.
  
  Say that we increase the intracellular concentration of potassium by
  10 mM, a +1 valence ion which contributes to POSITIVE membrane
  potential.


In your case K⁺ has a negative effect on the membrane potential, because of the high concentration gradient with reverse direction. So decreasing the concentration gradient will increase the membrane potential.

If you want to learn more about how to count membrane potentials, you should read this and its next section instead of wikipedia...