I have checked everything "homology of loop spaces"like, but was not able to find what is $H_*(\Omega^2S^3, \mathbb{Z})$. Therefore I ask you how to compute that?

2$\begingroup$ You know it mod p for all primes, and it's not hard to figure out the Bockstein. Then there's a theorem of Cohen that the only ptorsion you'll see will have exact order p, so that should be enough. If I have time later I'll try to write out what happens. $\endgroup$ – Dylan Wilson Sep 9 '17 at 23:02

$\begingroup$ I guess Cohen says that the $E_{\infty}$page (which as I said before is also the $E_2$page) of the Bockstein sseq will look like $\mathbb{F}_p[y]$ where $y=1$? (yeah it's weird that we have a polynomial algebra without graded commutativity, and I'm not quite sure I'm parsing his result properly.) $\endgroup$ – Dylan Wilson Sep 9 '17 at 23:34
There are homology isomorphisms $K(Br,1)\to \Omega^2_0 S^2$ and $\Omega^2 S^3\to \Omega^2_0 S^2$, so you are really asking about the homology of the stable braid group $Br$ (the colimit of the natural inclusions $Br_n\hookrightarrow Br_{n+1}$).
As expected there is no neat description with integral coefficients, but much is known. You'll find a nice summary in Section 4 of this paper of Vershinin .
The calculation you want is described in detail in Joe Neisendorfer's book Algebraic Methods in Unstable Homotopy Theory, Cambridge University Press, 2010. In particular, the EilenbergMoore spectral sequence collapses to give that $H_*(\Omega^2 S^3;\mathbb Z) = Cotor_*^{\mathbb Z[y]}(\mathbb Z, \mathbb Z)$. By calculating mod p, and then using the very simple Bockstein spectral sequence, he shows in Corollary 10.26.5, that $p$ annihilates the $p$torsion for all primes $p$. Thus $H_*(\Omega^2 S^3;\mathbb Z)_{(p)}$ is a graded $\mathbb Z/p$ vector space (above dimension 1) whose Poincare series could easily be worked out from $H_*(\Omega^2 S^3;\mathbb Z/p)$.