Finding The Number Of Inversions In A Permutation 
Let the be the following permutation:
  $(1 5 4)(3 6)\in S_6$

How do I count the number of inversions to calculate the sign of the permutation?

$(1 5 4)(3 6)=(1 5)(1 4),(3 6)=3$ so it has an odd sign

\begin{pmatrix}
    1 & 2 & 3 & 4 & 5 & 6 \\
    5 & 2 & 6 & 1 &  4  & 3
  \end{pmatrix}
before $5$ there are $0$ elements.

before $2$ there are $1$ elements (5).

before $6$ there are $0$ elements.

before $1$ there are $3$ elements (5,2,6).

before $4$ there are $2$ elements (5,6).

before $3$ there are $3$ elements (5,6,4).

Therefore there are $9$ inversions so it is $(-1)^9=-1$
Are both of the ways are ok? 

You have not done anything wrong except in thinking the permutation is even. There are indeed 9 inversions as you have calculated.
