As already observed above, if you consider a warped product metric $g_1 + f \cdot g_2$ on $S^2\times S^2$ obtained from a positively curved metric on each factor, it **will not have positive curvature**. This can be seen in the following way. The formula for the sectional curvature of "mixed planes" (i.e., spanned by vectors $X$ and $Y$, one from each factor) with respect to the warped metric is essentially (up to renormalization) the hessian of $f$ in the direction of $Y$, $\mathrm{Hess} \; f(Y,Y)$. Since the domain of $f$ is $S^2$ (which is compact), $f$ will have a minimum and a maximum, so its **hessian cannot be always positive (or negative) definite**. Thus, the curvature of mixed planes with respect to warped product metrics cannot be always $>0$ (or $<0$). You can find a slightly more precise description of this fact on a paper by Leysen and Verstraelen "On warped products and a conjecture of H. Hopf." Soochow J. Math. 13 (1987), no. 2, 175–178.

For a similar reason, a double-warped metric will also not have positive curvature, since the hessians of the warping functions are not definite. The Hopf conjecture is a nasty (but terribly intriguing) problem...

EDIT (due to Anton's comment): One comment above mentioned using Hsiang-Kleiner's result (a positively curved metric on $S^2\times S^2$ cannot have an isometric circle action) to answer the question. This indeed works if we are warping a positively curved metric on one $S^2$ that **has a circle in its isometry group** (e.g. the round metric) with another positively curved metric on the other $S^2$ (and the warping function is defined on this second factor). In this way, there is a circle acting isometrically in the warped metric and Hsiang-Kleiner's result applies. Nevertheless, in general, a positively curved metric on $S^2$ does not have an isometric circle action, so this reasoning cannot be used.