Mathematica is a term rewriting system and there are various ways to suppress automatic evaluation of expressions. The system doesn't evaluate Zeta for even arguments (see e.g. Zeta[Range[2, 20]]). Thus one of possible ways would be e.g.

Integrate[Log[1 - x]^5/x^5, {x, 0, 1}] /. 
 Times[x_, Pi^n_Integer] :> x Pi^n Inactivate[Zeta[n]]/Zeta[n] //
 TraditionalForm

Another way is to use HoldForm[Zeta[n]] instead of Inactivate[Zeta[n]] however the latter is more handy since you can use Activate then to evaluate the expression, e.g.

Activate[%]

-((5 Pi^2)/6) - (11 Pi^4)/18 - 30 (Zeta[3] + Zeta[5])

Mathematica is a term rewriting system and there are various ways to suppress automatic evaluation of expressions. The system doesn't evaluate Zeta for odd integer arguments (see e.g. Zeta[Range[2, 20]]). Zeta[n] yields expressions involving n-th powers of Pi, thus one of possible ways to achieve the goal would be e.g.

Integrate[Log[1 - x]^5/x^5, {x, 0, 1}] /. 
 Times[x_, Pi^n_Integer] :> x Pi^n Inactivate[Zeta[n]]/Zeta[n] //
 TraditionalForm

Another way is to use HoldForm[Zeta[n]] instead of Inactivate[Zeta[n]] however the latter is more universal and handy. You can use Activate (also with appropriate patterns) to evaluate the expression, e.g.

Activate[%]

-((5 Pi^2)/6) - (11 Pi^4)/18 - 30 (Zeta[3] + Zeta[5])

Nevertheless using such a replacement should be appropriately restricted to avoid possible ambiguities with expressions involving symbolic results in terms of powers of Pi.