The coefficient of friction should in the majority of cases, remain constant no matter what your normal force is.

In other words, the normal force is the force pushing the two surfaces together, and the stronger the normal force, the stronger the force due to friction.

Here's another way to think about it: because the force of friction is equal to the normal force times the coefficient of friction, we expect (in theory) an increase in friction when the …

Pressure is force divided by area.

In this lab, students will measure coefficients of friction and show that the frictional force doesn't depend on surface area in contact. With a given area of contact, increasing the normal force will pack the snow, decreasing the effective coefficient of friction. Since pressure equals force divided by the area of contact, it works out that the increase in friction generating area is exactly offset by the reduction in pressure; the resulting frictional forces, then, are dependent only on the frictional coefficient of the materials and the FORCE holding them together. The friction per area depends on the pressure between the surfaces …*the harder they're pressed together, the more friction you'd expect. When you apply a greater normal force, the frictional force increases, and your coefficient of friction stays the same.

The normal force isn’t necessarily equal to the force due to gravity; it’s the force perpendicular to the surface an object is sliding on. Halve the area, and the pressure is halved, so the friction per area is halved, so the total friction is still the same.
While exceptions such as this are easily found, the assumption that friction is proportional to the normal force is still reasonably valid in many cases and forms a useful model for many circumstances.