Wenn ich ein Kätzchen bekäme, würde ich mich sehr gut um es kümmern.
oder
Wenn ich ein Kätzchen bekommen würde, würde ich mich sehr gut um es kümmern.
Which sentence is right? I have read somewhere that most germans use the second formation most of the time. Is that right? Can I do that as well?
You can certainly use the second form in the sense that you will be understood. Even so, most people could probably say that they consider the first form more correct (even if they don't use it).
In practice, usage depends a lot on the type of verb you're using. With a full verb like 'bekommen', people tend to avoid the inflected forms of Konjunktiv to the point that they sound quaint. With shorter verbs and particularly auxiliary verbs the situation is reversed; people generally say "Wenn ich ein Kätzchen hätte", not "haben würde".
You can really use both without any problem. The "real" inflected Konjunktiv II form in your first variant
Wenn ich ein Kätzchen bekäme, würde ich mich sehr gut um es kümmern.
is on a higher level of language, and you would probably slightly prefer it in written language, especially in your example, where it also avoids the double "würde".
The main clause in your sentence is also a good example of a case where the form with "würde" is always preferable, namely when the inflected Konjunktiv II form is indistinguishable from the Präteritum form:
[...], kümmerte ich mich sehr gut darum.
This would be unusual nowadays, the "würde" form is much more common.
