Fermion as a mixture of particle and antiparticle The solution to the Dirac equation (in the Dirac basis) are 4 coupled fields. The first 2 of them represent a particle (spin up/down), the other 2 fields are the antiparticle (spin up/down). When the particle is observed from its rest reference frame, the antiparticle solutions are zero. However once the particle is moving, all the 4 fields become coupled.
Does it mean that a moving electron is a little bit of a positron at the same time (in the given reference frame)?
 A: Literally speaking the answer is negative. The charge of the state has to be always defined in view of the charge superselection rule. Thus for a particle described by Dirac equation, there are no things like coherent superpositions of electron states and positron states. A Dirac particle always stays  in a quantum state which is proper of  an electron or a positron, but never in a superposition of them. 
Nevertheless the quantum field associated to these particle describes both  particles and anti particles. 
A: Strictly speaking, a "particle" is only a quantum notion which must be understood in the context of quantum field theory, as a asymptotic state "in" or "out", in some interaction.
So, classicaly, stictly speaking, there is no "particle"
You are speaking of the classical Dirac equation, which is a classical field equation.
The quantum field version of the Dirac equation, is an equation, where the fields are operators, and these operators apply on states. The "particles" are only some particular asymptotic states, in interactions, being on mass-shell. The field operators are mixing creation operators of particles, and destruction operators of anti-particles, because you cannot separate the two. So the classical version of the Dirac equation, which is a approximate view of the quantum version of the Dirac equation, is mixing too the degrees of freedom of pseudo-particles and  pseudo-anti-particles. But keep in mind that only a quantum treatment is correct, where particles are states, and fields are operators.
