My situation is:

Player A plays Liber Mortis to his hero unit, and activates Liber Mortis action

Player B cuts-in, and plays Pilgrimage in respond, targets Player A's Hero unit

Action chain: 1. Pilgrimage 2. Liber Morits

My Questions are:

1.

After the effect of Pilgrimage (Player A's hero unit leaves play), how to resolve the action of Player A's Liber Mortis?

The target units of Liber Mortis are units in discard pile, so we cant terminate Liber Mortis action by making its action target leaving play. However, with the former part of this action "Corrupt attached Hero", can we stop its action by making this Hero unit leaving play? Is it a conditional action?

2.

The corrupt behavior of Liber Mortis is seen as cost of its action, or just part of its action?

Thank you for attantion. Sorry for my poor English

If you play the pilgrimage in respond to him playing Liber Mortis, his target will be gone, and Liber Mortis will fizzle.

If you wait for Liber Mortis to resolve and get attached to the Hero (I don't see why you want to wait for that before playing pilgrimage?), he can still active Liber Mortis' action, since the corruption of the Hero is a cost (do X to do Y). The Hero will be corrupted before the action chain starts to resolve.