In the case of alternating current, the current density drops exponentially with distance from the outer surface of the wire (the "skin effect"), as explained by Martin Beckett. This can be shown analytically from the quasistatic approximation to Maxwell's equations, as is done in Jackson chapter 5.
The case of direct current is more interesting. First, you need to specify the external electric field ${\bf E}_0$ that "pushes" the current. This is usually taken to be uniform and parallel to the wire. The currents through the wire tend to attract each other and therefore cluster together (known as the "pinch effect"). The DC pinch effect is discussed in http://aapt.scitation.org/doi/abs/10.1119/1.1974305, http://aapt.scitation.org/doi/abs/10.1119/1.14075, and http://aapt.scitation.org/doi/abs/10.1119/1.17271. It turns out that Maxwell's equations are not enough to uniquely determine the current density distribution through the wire's cross-section; you also need to specify a microscopic model for the charge carriers.
At one extreme, you can treat both the positive and negative charge carriers as completely mobile and with equal charge-to-mass ratios. This is a good description of current conduction through plasmas, and plasma pinches can be strong enough to crush metal.
At the other extreme, you can treat the positive charges as completely stationary in the lab frame, at fixed density, and "immune" to the electromagnetic fields, with the current due entirely to the motion of the mobile negative charge carriers. This is a more realistic model for a metal wire, since the interatomic and Fermi exchange forces between copper atoms are much, much stronger than those induced by typical applied fields and electron currents. It turns out that in the lab frame, the wire's total linear charge density must be zero at equilibrium (otherwise it would exchange electrons with the fixed sources and sinks at the battery until it neutralized), but in the rest frame of the moving electrons, the bulk volume charge density must be zero (otherwise the electrons would experience a radial electric force drawing them toward or away from the wire's axis).
Combining these requirements, you get the following picture: define $R$ to be the wire's radius, $\rho_0$ to be the density of positive ions in the lab frame (in which they are at rest), $\beta = v/c$, where $v$ is the electron's drift velocity as seen in the lab frame, and $\gamma = 1/\sqrt{1-\beta^2}$. In the lab frame, the bulk positive volume charge density is $\rho_0$ and the bulk negative volume charge density is $-\gamma^2 \rho_0$, which is greater in magnitude. So the bulk net volume charge density $(1 - \gamma^2)\rho_0 = -\beta^2 \gamma^2 \rho_0$ is negative, and there is a radially inward electric field whose magnitude increases linearly with radius. (The internal generation of this radial electric field is sometimes called the "self-induced Hall effect.") The electric field balances out the radially inward attraction between electrons due to the current flow. There is a compensating positive surface charge density $\sigma = (R / 2) \beta^2 \gamma^2 \rho_0$ around the surface of the wire which balances out the negative bulk volume charge, so the radial electric field vanishes outside the wire. This surface charge is at rest in the lab frame, so it does not contribute to the current.
In the electrons' frame, there is no bulk volume charge density or radial electric field inside the wire. (There is a magnetic field from the motion of the positive ions, but the electrons don't feel it since they are at rest in this frame.) The surface charge in this frame is $\sigma' = (R / 2) \beta^2 \gamma^3 \rho_0$, and the total linear density in this frame is $\lambda' = 2 \pi R \sigma' = \pi R^2 \beta^2 \gamma^3 \rho_0$. In this frame, there is a radial electric field outside the wire, which does not effect the electons, but does attract or repel charged particles outside the wire.
But in a copper wire with typical currents, the electrons are extremely nonrelativistic ($\beta \ll 1$), so the net negative bulk charge and positive surface charge are extremely tiny.
