When discussing relativistic quantum mechanics, the single particle picture is dropped completely
Not completely. You can still consider 0-particle and 1-particle states. The 0-particle state is usually called the vacuum, and can be denoted by a ket like $|\Psi^{(0)}\rangle$, which is annihilated by all the particle annihilation operators:
$$
c_i |\Psi^{(0)}\rangle = 0\;.
$$
1-particle states look like:
$$
|\psi^{(1)}_i> = c^\dagger_i|\Psi_0\rangle
$$
2-particle states look like:
$$
|\psi^{(2)}_{ij}> = c^\dagger_j c^\dagger_i|\Psi_0\rangle\;,
$$
where the nice thing about the QFT formulation is that the bosonic or fermionic nature (the wave function symmetry) is explicit via the operator relations.
which makes the description of just two entangled particles difficult.
No more difficult than anything else really. The difficulty probably depends on how familiar with QFT one is.
However can some description be built from Dirac's equation or quantum field theory?
Yes, some description can be built. It is probably not helpful, though, in practice, since quantum computers are non-relativistic condensed matter systems. A familiarity with the Dirac equation can be helpful if the quantum computer is built, say, from Majorona fermions that emerge out of some types of condensed matter band structure.