First, let us determine the relations.
We know that 3 of the 4 male members (the remaining 4th being W) are brothers of each other (otherwise they could not have 2 brothers each). One of them is T (we know that T is male because he is U's brother), and now we know that U is the only sister of the three. Who are T's other 2 brothers?
Obviously neither of U, V and Q, since all of them are female. S is one of them since both remaining male members of the family (apart from T and W) should be T's brothers. But R cannot be T's brother since S is brother-in-law (not the blood brother) of R. So, the 3rd (remaining) brother is P.
So, P, S and T are brothers, and U is their only sister. Since S is the brother-in-law of R, then R (being female) must be married to one of the S's brothers. It must be P since T is unmarried.
Since V is the mother-in-law of R, she must be the mother of P,S,T and U. U is Q's sister-in-law, that is, Q is married to U's brother, namely S (since T is unmarried, and P is married to R). Q is W's daughter-in-law, so W must be V's husband.
Final tree (sorry for the crude drawing, males are squares, females are circles):
Now, let us proceed to the newspapers. First, we know that R reads India Today (H) (since V is R's and Q's mother-in-law, and Q reads other newspaper), Q reads Outlook (E), and T reads Frontline. S does not read Businessworld, Sportsstar or The Week, so he must read either India Today (R) or Outlook (H). But S is Q's husband who also reads Outlook, so S must read India Today (R).
The Week and Outlook (H) are read by male members, that is, P and W (since S and T read something other). P does not read The Week, so he reads Outlook (H), and W reads The Week. U does not read Businessworld, so she must read Sportsstar, the only remaining newspaper. And Businessworld should remain to V.
Final answer (apart from the family tree):
P reads Outlook (H),
Q reads Outlook (E),
R reads India Today (H),
S reads India Today (R),
T reads Frontline,
U reads Sportsstar,
V reads Businessworld,
W reads The Week.
The clause "V is the mother-in-law of R and who reads India Today (H)" is interpreted as if R and one who reads India Today (H) could be the same person (and in fact, it is). If we assume that these must be different persons, we get a contradiction.