Gravitons are the application of the methods of quantum field theory to the *linearized* version of general relativity (that is, a kind of weak-field limit). When you take general relativity, and you apply a series development of the metric and limit yourself to first order, you find the linearised Einstein equations.
For short: absence of gravity implies that the metric is the Minkowski metric of special relativity (which can be seen as a diagonal matrix with elements (1,-1,-1,-1) on the diagonal in an orthonormal coordinate system). Gravity manifests itself by a deviation of the metric from this Minkowski (or "flat") metric, and the Einstein equations give you the link with the matter and energy distribution. In a certain way, the Einstein equations are to the metric, what the Maxwell equations are to the E and B fields: the Maxwell equations give you the link between the E and B fields and the distribution (and motion) of charges. The Einstein equations do the same for the metric and the distribution (and motion) of matter and energy.
However, there's a big caveat: the Einstein equations are non-linear, as where the Maxwell equations are linear. This means that if you double all charges and currents, you will double the E and B fields. Not so for gravity. This is why Einstein's theory is so rich in strange solutions (and so difficult to handle).
But, for WEAK gravity, we know that the gravitational response is linear (twice the mass gives you twice the gravitational pull in Newton's theory of gravity). So if we do a series expansion of the metric tensor around the flat value (Minkowski tensor), and limit ourselves to small deviations and first order, we can linearise Einstein's equations, and we get something that has some similitude to the Maxwell equations...
Except that our thing is now a second-order tensor, and not a (first-order) vector (E-field, say).
These are the linearised Einstein equations. If you push it somewhat, you can get out Newtonian gravity (a bit like you can get out Coulomb electrostatics out of Maxwell's equations). But you get also some wave equations, similar to the equations of electromagnetic waves.
And if you apply quantum field theory bluntly to those wave equations, well, you get a quantum field theory that looks somewhat like quantum electrodynamics. Except that you get a different "mediating particle", not a photon, but a "graviton". Because the field quantity is a 2nd order tensor (the metric tensor, or better, the deviation of the metric tensor from the flat metric tensor), our "graviton" will be a spin-2 particle. Because in EM, the field is a vector (say, E-field), the photon is a spin-1 particle.
So, "by imitation" of what happens to the quantum version of electromagnetism, one can quantize the linearised version of Einstein's equations, and one then finds a particle, called the "graviton".
Except that the theory is full of difficulties. There where a trick in electromagnetism, called, renormalization, allows one to get rid of all infinities that devellop during the calculations, this trick doesn't work for the graviton. And we remain with the difficulty that we patched quantum theory upon a *linearised* version of Einstein's equations. So it is not really clear what the graviton is supposed to mean.
Now, there's a lot more to say about this, but that's beyond my own knowledge.
As to "gravitational self-coupling", well, because of this linearised version of the equations on which one applied quantum field theory, there is no direct graviton-graviton interaction, as there is also no direct photon-photon interaction (there are indirect interactions, through pair creation, but this is even out of scope for gravitons, as higher-order diagrams diverge because of the lack of renormalisation).
I think that for the same reason, there are no direct photon-graviton interactions, but I might be wrong here.
But self-particle interactions ARE considered in other theories. Non-abelian gauge theories, such as QCD, do have gluon-gluon interactions. That is because the equations one starts with are non-linear (but the gauge stuff saves us there).
Does the graviton have "its own gravitational field" ? Well, because we linearised that away, no. Like the photon doesn't have its own charge (but there it is believed to be exact).
