The Hurewicz theorem tells us that if $X$ is a path-connected space then $H_1(X, \, \mathbb{Z})$ is isomorphic to the abelianisation of $\pi_1(X)$. This gives a potential method for computing the abelianisation of a (sufficiently nice) group $G$: realise it as the fundamental group of a space $X$ and then compute $H_1(X, \, \mathbb{Z})$ by your favourite means.
Is this method ever used in practice? Are there nice examples of abelianisations which are easily (best?) computed in this way?
Yes, the fundamental group of the Hawaiian earring $\pi_1(\mathbb{H},b_0)$ is an important group which is sometimes called the free sigma product $\#_{\mathbb{N}}\mathbb{Z}$. Its is often defined in purely algebraic terms as a group of "transfinite words" in countably many letters. In many ways this group behaves like the non-abelian version of the Specker group $\prod_{\mathbb{N}}\mathbb{Z}$ and it is the key to the homotopy classification of 1-dimensional Peano continua. However, it's abelianization is not $\prod_{\mathbb{N}}\mathbb{Z}$; it is a good deal more complicated. The abelianization was first described by Eda and Kawamura in the following paper:
The Singular Homology Group of the Hawaiian Earring, Journal of the London Mathematical Society, 62 no. 1 (2000) 305–310.
The proof explicitly uses the Hawaiian earring and the authors mention in Remark 2.7 that
Though the question itself is formulated algebraically, we have not succeeded in finding a purely algebraic proof of it.
One way to represent the isomorphism class of this abelianization is as $$Ab(\#_{\mathbb{N}}\mathbb{Z})\cong \prod_{\mathbb{N}}\mathbb{Z}\oplus \prod_{\mathbb{N}}\mathbb{Z}/\bigoplus_{\mathbb{N}}\mathbb{Z}$$
Similar methods have been used to identify abelienizations of fundamental groups of other 1-dimensional spaces, e.g. the Menger curve.
A usually hard problem in Algebraic Geometry is computing the fundamental group of the complement $U=\mathbb{P}^n - V$, where $V$ is a reduced hypersurface.
At least in principle, the computation of $\pi_1(U)$ can be carried out by using Seifert-Van Kampen theorem; the tricky part is that the relations in the presentation depend not only on the singularities of $V$, but also on their mutual position.
A classical example due to Zariski is when $V \subset \mathbb{P}^2$ is a curve of degree $6$ having six ordinary cusps and no other singularities. Then $\pi_1(U)$ is the free product $\mathbb{Z}/2 \mathbb{Z} * \mathbb{Z}/3 \mathbb{Z}$ if the six cusps lie on the same conic, and the direct product $\mathbb{Z}/ 2 \mathbb{Z} \times \mathbb{Z}/3 \mathbb{Z}$ otherwise.
However, by using Lefschetz duality we can compute in a straightforward way $H_1(U, \, \mathbb{Z})$ for all hypersurfaces, so that the abelianization of $\pi_1(U)$ is always known even if a presentation for $\pi_1(U)$ is not available. In fact, there is the following result that can be found in A. Dimca's book Singularities and topology of hypersurfaces, Chapter 4.
Proposition. Assume that the hypersurface $V \subset \mathbb{P}^n$ has $k$ irreducible components $V_1, \ldots, V_k$ of degrees $d_1, \ldots, d_k$. Let $d$ be the greatest common divisor of the integers $d_1, \ldots, d_k$. Then $$H_1(U, \, \mathbb{Z}) = \mathbb{Z}^{k-1} \oplus \mathbb{Z}/ d \mathbb{Z}.$$
In particular, $H_1(U)$ does not depend on the singularities of $V$, but only on the number and degrees of its irreducible components. As a consequence, we recover the fact that in both cases of Zariski's example quoted above (where $k=1, \, d_1=6$) the abelianization of $\pi_1(U)$ is the cyclic group of order $6$.
The mapping class group of a smooth manifold $M$ is the group of all its self diffeomorphisms up to isotopy, i.e. $\pi_0(\operatorname{Diff}(M))\cong \pi_1(B\operatorname{Diff}(M))$.
A large portion of geometric topology is concerned with gaining a better understanding of the homotopy type of $B\operatorname{Diff}(M)$, in particular of its cohomology as the latter is the ring of characteristic class of smooth $M$ bundles.
In the last 15 years, initiated by Madsen and Weiss' solution of the Mumford conjecture, the understanding of the homology of $B\operatorname{Diff}(M)$ has fundamentally improved which led, among others, to calculations of $H_1(B\operatorname{Diff}(M))\cong \pi_0(\operatorname{Diff}(M))^{ab}$ for some $M$, e.g. by Galatius and Randal-Williams in the following paper.
Abelian quotients of mapping class groups of highly connected manifolds, Mathematische Annalen June 2016, Volume 365, Issue 1, pp 857–879
