How do charged particles interact with each other? As we know, charged particles have polarity and like charges repel each other and unlike charges attract each other, and we have Coulomb's law to find that force. But how does it work? Does it work like gravity, like when two like particles are apart do they still repel? Is there any observational data relating to it?
 A: We have frameworks in physics where questions can be answered: the classical electrodynamics framework, the quantum framework, the special and general relativity framework etc. Frameworks differ in the variable's range of validity, but blend smoothly in the overlap region.
In current day physics particle is a name given to electrons, muons etc, in the particle data table of the standard model of particle physics.. This is the quantum mechanics framework.
Your question is in the classical electromagnetism framework, so these are classically defined particles carrying charge .

As we know charged particles have polarity

yes, if by polarity you mean + or  - charges.

and like charges repel each other and unlike charge attract each other, and we have coulomb's law to find that force. 

Correct.

But how does it work, does it work like gravity, like when two like particles are apart do they still repel,

Yes, the field of each particle  overlapping generates a repulsive force, following Coulomb's law, theoretically no matter how far apart they are.
Note the term law. Laws are axioms in a physics theory, they tie measurements and observation to the mathematical model, in this case the 1/r^2 law.

is there any observational data relating to it.

A lot. Classical electrodynamics is a theory with Maxwell's equations that describes and predicts all possible situation macroscopically, and has not been falsified by measurements. This includes Coulomb's law.
Now let us go back to particles in the microscopic, quantum mechanical framework, like two electrons repelling each other :

The mathematical formulae represented by this feynman graph, for large distances will display Coulomb law behavior. At small distances it  is complicated and needs the study of new mathematical tools as displayed by the graph above. 
