Expand (1+y+ky^2)^8 until y^3
You could do the tedious task of expanding and then collecting powers of y^3.
I let Wolfram do this for me
http://www.wolframalpha.com/input/?i=expand+%281%2By%2Bky%5E2%29%5E7
It might be easier to read the expansion in reverse order
1 + 7y + 21y^2 + ...
Notice that y^3 appears only twice, as
35y^3 and 42ky^3
I don't like the wording of your question. As it appears, we would need the full expansion.
You can see the leading terms changing by changing
the exponent in the yellow window of the Wolfram page from 2 to 3, to 4 , etc
A more logical question would have been to find the coefficient of the y^3 term,
which would have been (42k + 35)y^3
I came across this very interesting analysis of raising a trinomial to different powers, which I had never seen before.
Very neat pattern, let's you do any power.
http://www.qc.edu.hk/math/Advanced%20Level/Trinomials.htm