Since has 7 conjugacy classes, it has 7 irreducible characters.

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Size 1 10 20 30 24 15 20

Class 1, (12), (123), (1234), (12345), (12)(34), (12345)

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X1 1 1 1 1 1 1 1

X2 1 -1 1 -1 1 1 -1

X3 4 2 1 0 -1 0 -1

X4 4 -2 1 0 -1 0 1

X5 6 0 0 0 1 -2 0

X6 5 1 -1 -1 0 1 1

X7 5 -1 -1 1 0 1 -1

X1 is the character for the trivial representation of S_5.

X2 is the character for the alternating(sign) representation of S_5.

X3 is the standard representation of S_5.

X4 = X2 X3. It turns out to be an irreducible character of S_5.

X5 = 1/2(X3(g)^2 - X3(g^2)),

X_6,

X_7 = X2 X5.

These characters satisfy both "row orthogonality theorem" and "column orthogonal theorem" (link).

Now you need to do some calculations and fill the table.