Allow me the excuse to mention a cute way of constructing U_{q}(sl_{2}) as a q-analogue. Presumably this is ahistorical, since I suspect that initially U_{q}(sl_{2}) was constructed first over the formal disc, then later it was realised that it was definable over $\mathbb{Q}(q)$.

Anyhow, let S be a finite set. Consider a vector space with basis $\{[I]\mid I\subset S\}$. Define operators e,f,h on this vector space by
$$e[I]=\sum_{i\notin I}[I\cup\{i\}]$$
$$f[I]=\sum_{i\in I}[I\setminus \{i\}]$$
$$h[I]=(2|I|-|S|)[I].$$

You can check that this is a representation of sl_{2} (it's just a tensor power of the standard represenation in disguise).

Now q-analogise! How? Well we should replace the idea of a subset of a set with that of a subspace of a vector space (of course working over the finite field with q elements).

Let V be a finite dimensional vector space over $\mathbb{F}_q$. We should consider the space of all complex-valued functions on the Grassmannian Gr(V) of all subspaces of V. Well actually we'd really like something of dimension 2^n, so we should consider only those functions invariant under the action of a Borel subgroup.

Let F be the space of pairs (W,W') with $W\subset W'$ two subspaces of V satifying dim(W')=dim(W)+1. There are two obvious maps $p_1,p_2:F\to Gr(V)$.

Now we can define two operators on our space of functions on Gr(V), namely $E=(p_2)_!(p_1)^!$ and $F=(p_1)_!(p_2)^!$. Here p^{!} is pullback of functions and p_{!} means sum the function over the fibre. These are almost the generators of the quantum group we're after, to make the formulae match up nicely, we need to modify the definition of p_{!} by multiplying it by a factor of q^{-d/2} where d is the dimension of the fibre summed over (a Tate twist).

Now you have two operators E and F, and when you compute their commutator, you naturally find another operator K such that $$EF-FE=\frac{K-K^{-1}}{q^{1/2}-q^{-1/2}}.$$
Computing KE=qEK and qKF=FK is now easy.

But even better, this approach gives more than just the product structure on the quantum group. You can also get the coproduct. I will be brief since I haven't checked all powers that q that appear. I haven't thought about obtaining the antipode.

Consider a short exact sequence of vector spaces 0-->V'-->V-->V''-->0. Given functions f' on Gr(V') and f'' on Gr(V''), one can define a function on Gr(V) by
$$(f'\otimes f'')(X)=f'(X\cap V')f''(X/(X\cap V')).$$

There is a reasonable chance this needs correction by a power of q. The point however is that this defines an isomorphism between Borel-equivariant functions on Gr(V) and the tensor product of Borel-equivariant functions on Gr(V') with Borel-equivariant functions on Gr(V''). (If you drop Borel-equivariant, then you're still getting a map, you're just not deforming a representation of sl_{2}).

I leave it as an exercise to the interested reader to use this to deduce the coproduct formulae in U_{q}(sl_{2}).