It seems that most people draw $\ce{ClO4^{-}}$ with three double bonds and the negative charge on the oxygen: http://www.chemspider.com/Chemical-Structure.109953.html
However, it would seem that it is more stable for the negative charge to be on the chlorine as it is more electronegative? (thus giving 4 double bonds to each oxygen)
You have to be careful to distinguish between the formal structures we draw to count electrons and the actual structure of the molecule. In reality the 4 bonds are the same in the perchlorate ion. But when we are counting electrons we find it bothersome to put fractional charges on atoms because that makes adding up harder. And we want to have single or double bonds for the same reason.
So we draw complete bonds which have two electrons each and balance out the total number of electrons with charge. Moreover the charge should be on the more electronegative atom, hence it goes on oxygen. However, this formalism gives an incomplete picture of the true molecular orbitals that make up the bonding (crudely, nature isn't as fussy as us about doing calculations with fractions and is happy distributing that negative charge over all the oxygens).
A slightly more sophisticated version of our whole-number accounting formalism recognises this via an extension of the simple formalism to include "resonance". Here we pretend that a number of structures "resonate" to give the true structure. In this case we can draw the single bond and negative charge on each of the four oxygens and pretend that each of those structures contributes equally to the true molecule structure.
The idea of resonance is often used to bridge the simple accounting model where bonds have two electrons, with real world structures where the electrons and bonds are spread across several atoms.
I think the best argument comes from resonance.
Drawing perchlorate with three $\ce{Cl-O}$ double bonds and the negative charge on the oxygen allows for three resonance structures to be drawn (four total). This resonance stabilization dominates the Lewis structure over simply drawing four double bonds to chlorine.
In other words, energetic benefit of charge spread out over four oxygens > charge localized on one chlorine.
