Cortez Chip states: "trash" Choose a piece of ice. The Corp must pay 2 (bits) as an additional cost to rez that ice until end of turn.

Forged Activation Orders states: Choose an unrezzed piece of ice. The Corp must either rez that ice or trash it.

Can I play a Forged, the Corp then chooses to Rez targeted ice and then in response I trash Cortez Chip to increase the cost?

The timing for triggering Paid Abilities is after resolving an action (or at the beginning of your turn), so you will have to trigger the Chip before playing Forged Activation Orders. Otherwise, you will have to wait for the playing of the Orders completely resolved (i.e. either the Corp rez the ICE or trash it), then you will have opportunity to trigger the Chip's effect.

When you trash Cortez Chip, it's ability lasts until the end of the turn. As said above, do this first.