It's a theorem of Garland that $K_2(R)$ is finite. Perhaps the best way to get a handle on it is to use Quillen's localization sequence $$0\rightarrow K_2(R)\rightarrow K_2(F)\stackrel{T}{\rightarrow} \oplus_v k(v)^*\rightarrow 0,$$ where $F$ is the fraction field and the $k(v)$ are the residue fields. The map $T$ is the sum of the tame symbols, which is surjective by a theorem of Matsumoto. The injectivity on the left follows from the vanishing of $K_2$ for finite fields.
This isn't much of an answer, but considering $K_2(R)$ as a subgroup of $K_2(F)$ seems a reasonable way to start some concrete considerations. For a detailed discussion of an algorithm that proceeds essentially along these lines ('Tate's method), see the paper
Belabas, Karim; Gangl, Herbert Generators and relations for $K_2( O_F)$. $K$-Theory 31 (2004), no. 3, 195--231.
Added, 8 July:
I'm sure most people know this, but I forgot to mention (for newcomers) the fact that $$K_2(F) = F^\times\otimes_{\mathbf Z} F^\times/\langle a\otimes(1-a)\mid a\not=0,1\rangle,$$ which I suppose motivates the original question, and makes it worthwhile to view $K_2(R)$ as a subgroup.