When is there a ring structure on an abelian group $(A,+)$?

In a unitary ring you always have the fundamental concept of "characteristic". If you have a torsion Abelian group $G$, and you want a unitary ring structure on $G$, then first of all you need to choose a unit. If $g\in G$ is such a unit, then you are forced to have a ring of characteristic $n=|g\mathbb Z|$. This tells you that $nh=0$ for all $h\in G$, so if $G$ does not have a finite exponent, you cannot have a ring structure on $G$.

If you want a source for many torsion-free counterexamples you can proceed as follows. You can use a famous theorem of Saharon Shelah to find, for any infinite cardinal $\alpha$, $2^\alpha$ many non-isomorphic torsion-free abelian groups of cardinality $\alpha$ and whose endomorphism ring is a subring of the rationals. In particular, given such a $G$, $\operatorname{End}(G)$ is countable. Notice that, if you have a ring structure on $G$, then there is an injection $G\to\operatorname{End}(G)$ sending $g$ to left-multiplication by $g$.

Let me add that these matters are nicely discussed in Chapter XVII of the second volume of Infinite Abelian Groups by Laszlo Fuchs (the title of the chapter is "Additive groups of rings").

EDIT: I briefly sum up some of the matters treated by Fuchs. Let me start by quoting his final "note".

The problem of defining ring structures on an additive group was raised by Beaumont who considered rings on direct sums of cyclic groups. Nearly at the same time, Szele investigated zero-rings, and Ridei and Szele and Beaumont and Zuckerman described the rings on subgroups of the rationals. A rather systematic study of constructing rings on a group appeared in Fuchs where the fundamental role of the basic subgroup was pointed out. More satisfactory results have been obtained for torsion-free groups of finite rank by Bedumont and Pierce [...].

It would be a serious mistake to expect too much from a study of the additive structures of rings, as far as ring theory is concerned. In many important cases the additive structures are too trivial [e.g., torsion-free divisible or an elementary $p$-group] to give any real information about the ring structure. This especially applies to the torsion-free case, where a close interrelation between the additive and the multiplicative structures can be expected only if the additive group is more complicated. One should, however, remember that there are intriguing questions even if the additive group is too easy to describe; for instance, we do not know of any uncountable Noetherian ring whose additive group is free.

Let me say that your question is open in general but there are answers for particular classes of groups, such as torsion-free groups of rank one, where one can not only say when a ring structure exists but also encode somehow all the possible ring structure, even non-associative or non-unitary. The fact is that one of the tools for studying ring structures on a group $G$ is the group of multiplications on $G$, $\operatorname{Mult}(G)$. This is a group in general only if you include non-associative multiplications.

The chapter starts studying general properties of $\operatorname{Mult}(G)$ (for example one can show that $\operatorname{Mult(G)}\cong\operatorname{Hom}(G\otimes G,G)$). After that it passes to general conditions on torsion and torsion-free groups (especially divisible, or finite rank, or even rank $1$ for the best results). Notice that the main focus is not exactly on your question but it is on an even more delicate matter, that is, given a group $G$ try to classify all the rings that have $G$ as underlying group. Proceeding with the chapter more particular questions are investigated. For example, one can give a very precise form for the additive groups of Artinian rings (see Theorem 122.4). There are also precise results for regular rings and the chapter ends (as all the chapters of this book) asking some intriguing questions on the matter.

It may happen that this classical book is not up to date as it goes back to 1973 but, in my experience, you can be quite sure that, if something related to Abelian groups was known before 1973, then it is in this book. I was looking for more recent information about your question something like 3 years ago but, as far as I remember, I could not find anything relevant after Fuchs' book.