I have heard of the fact that Wittgenstein rejected Cartesianism by the Private Language Argument. But the connections are not clear for me. Also I'd like to know if there is any other piece of writing by Wittgenstein that criticized Cartesianism.
There are several ways of answering this question - Wittgenstein never makes things easy. I offer the following.
Cartesian dualism - or just dualism - draws a distinction between mind and behaviour. The mind thinks or knows; the body acts. We don't need to go into the infinite varieties of behaviourism to see what Wittgenstein's basic thrust against dualism is.
On Wittgenstein's account in the 'Brown Book' and later in the 'Philosophical Investigations' is that mind and body are not distinct; there is a logical connection between mind and behaviour. ('Mind' is not a favoured term of Wittgenstein's; he is more apt to refer to 'the mental' but it does not affect the main point here.)
First thing to note is that for Wittgenstein it makes sense to ascribe mental phenomena only to beings that can exhibit such phenomena in their behaviour. 'Only of a living human being and what resembles (behaves like) a living human being can one say : it has sensations; it sees; is blind; hears; is deaf; is conscious or is unconscious' (PI, §281). There is no sense in saying that a plant feels pain since it has no way of manifesting pain in its behaviour.
Next up, our mental terms would have different meanings from any they now have if they were not tied to behavioural criteria. For instance if I say of my four-year-old child, 'She can read', I mean not simply that she has a certain mental capacity; I mean that she can behave in certain ways : pick up a newspaper, book or comic and read out the words (behaviour) or tell us what she has read (another form of behaviour). She might choose not to engage in the relevant behaviour but if she really could not do so, 'She can read' would only be true if we changed the meaning of 'read'. In the standard case the mental capacity to read implies some form of behaviour and not just the occurrence of certain private, non-behavioural occurrences 'in' the mind.
Useful guides for further reading : AC Grayling, 'Wittgenstein', Oxford Past Masters series, 1996, 85-6, 90-1, 94, 96, 97; and Hans-Johann Glock, 'A Wittgenstein Dictionary', Oxford : Blackwell, 1996, 55-8. esp. 57-8.