Phylactery Lich’s first ability doesn’t target the artifact. No player may take actions between the time you choose an artifact and the time you put a phylactery counter on it.

If you control no artifacts as Phylactery Lich enters the battlefield, its first ability won’t do anything. As soon as it enters the battlefield, its last ability will trigger (unless you control some other permanent with a phylactery counter on it) and you’ll have to sacrifice it.

If Phylactery Lich and an artifact are entering the battlefield under your control at the same time, you can’t put a phylactery counter on that artifact. You must choose an artifact you control that’s already on the battlefield.

Phylactery Lich’s last ability is a “state trigger.” Once a state trigger triggers, it won’t trigger again as long as the ability is on the stack. If the ability is countered and the trigger condition is still true, it will immediately trigger again.

Phylactery Lich’s last ability checks your permanents for any phylactery counters, not just the specific one that it caused you to put on an artifact. For example, if you control two Phylactery Liches and two different artifacts with phylactery counters, neither Phylactery Lich will be sacrificed if one of those artifacts leaves the battlefield.