When does artificial intelligence just become intelligence?

When we come to use the word "intelligence" simply to refer to a set of capacities, and when it becomes natural to treat non-biological machines and biological machines indifferently when assessing those capacities. See this week's NEW YORKER for a fascinating article on how that is playing out in the domain of chess. (Tom Mueller, "Your Move: Creating a Better Chess Player, NEW YORKER Dec 12, 2005, pp 62-69)

Does personality prove self-consciousness? If an animal has a defined and changeable personality (can get depressed, etc.), is it then self-aware?

There is a difference between consciousness and sel-consciousness. It is one thing to be aware of things around oneself, or even of one's own physical states, and another to be aware that one is so aware. The latter is self-consciousness. Personality is yet a third thing. As a reasonable first pass we could take it to be a bundle of behavioural, affective and cognitive dispositions. Those would seem to require consciousness, but not self-consciousness.

What, if anything, can you boil one's self down to, outside any notion of soul or essence?

Many philosophers, especially those in the Buddhist tradition (Nagasena, Candrakirti, Santideva,or see Hume for a Western sympathiser), have argued that there is nothing that one can "boil oneself down to," that is, that the self has no existence independent of convention. Others have argued that there is some basic subjective entity, either substantial (Descartes) or transcendental (Kant, Schopenhauer) that undergirds our identity.