(Note: I apologize in advance for the ugly typesetting, but it seems that the usual way of including inline equations is now rendering them on separate lines, at least in Firefox, and I couldn’t find a workaround by Googling!)
If you’ve studied calculus of variations before, it may help to use the notation \delta\mathbf{u} for the test function instead of \mathbf{v}\text{ ,} then think about setting the Gateaux derivative of \Pi with respect to \mathbf{u} in the direction \delta\mathbf{u} to zero (by analogy to seeking the minimum of a univariate function by looking for an argument such that its ordinary derivative is zero):
d\Pi(\mathbf{u};\delta\mathbf{u}) = \delta\Pi = \int_\Omega \frac{\partial\psi}{\partial\mathbf{F}}:\delta \mathbf{F}\,d\Omega - \int_{\partial\Omega} \mathbf{T}\cdot\delta\mathbf{u}\,d\partial\Omega = 0\text{ ,}
where the variation \delta\mathbf{F} is also a Gateaux derivative with respect to displacement of \mathbf{F} = \operatorname{grad}\mathbf{u} + \mathbf{I}\text{ ,} i.e.,
\delta\mathbf{F} = d\mathbf{F}(\mathbf{u};\delta\mathbf{u}) = \operatorname{grad}(\delta\mathbf{u})\text{ .}
Making the substitutions \partial\psi/\partial\mathbf{F}\to\mathbf{P}~,~\delta\mathbf{F}\to\operatorname{grad}(\delta\mathbf{u})~,~\delta\mathbf{u}\to\mathbf{v}\text{ ,} you recover the weak form
\int_\Omega \operatorname{grad}\mathbf{v}:\mathbf{P}\,d\Omega - \int_{\partial\Omega}\mathbf{T}\cdot\mathbf{v}\,d\partial\Omega = 0\quad\forall\mathbf{v}\text{ .}