To my mind, the overriding rules here are these:

Alternate Form

The creature retains the type and subtype of its original form.

Undead Type

• No Constitution score.

You cannot have an undead that has a Constitution score, so since Alternate Form has you retain your type, you cannot gain a Constitution score. I believe this is “more specific” than the general rule that you gain the target creature’s physical ability scores when using Alternate Form.

You could argue the opposite way, however, that Alternate Form is a more specific case than the general rule that undead do not have a Constitution score. But note that this results in a living (has a Constitution score) undead (retains that type), which to me is a good enough reason to side the other way.

As for using Charisma in place of Constitution, this is a common feature for undead, but not a universal one. In some places, that privilege requires a feat (e.g. Undead Meldshaping in Magic of Incarnum). Depending on your build, I might want to tax you for it; Cha-to-everything is a fairly potent strategy as it is, and undead is a desirable type anyway. Getting to use Charisma for yet more things, and avoiding the “weakness” of the undead type’s lack of Constitution, might be problematic. But if you weren’t already pursuing a Cha-to-everything build, I’d probably give it freely.

This is complicated, but it's unchanged by the 3.5 revision

Note: The effect of lowering a creature's Constitution ability score modifier is unchanged by the 3.5 revision. Likewise, the 3e quotations below were only barely changed—and not in any meaningful way—by the 3.5 revision (also see DMG (2003) 290 and here). This makes answers to this question also relevant. I've answered this question completely nonetheless in hopes that a different voice may clarify matters further.

The Dungeon Master's Guide (2000) on Ability Score Loss, in part, says

If a character’s Constitution score drops, then he loses 1 hit point per Hit Die for every point by which his Constitution modifier drops. For example, at 7th level, Tordek is hit by poison that causes his Constitution to drop from 16 to 13. His bonus falls from +3 to +1, so he loses 14 hit points (2 per level). A minute later, the poison deals another 8 points of temporary Constitution damage, dropping his score to 5 and his modifier from +1 to –3. He loses another 28 hit points—for a total of 42 hit points lost because of an overall 6-point drop in his Constitution modifier. (72)

It's the examples that make this really go. The DMG is, I think, trying here to simplify the process of reducing a creature's Constitution score, but, while doing so, it makes the process less clear and more complicated than it is. The DMG suggests that a creature treat its Constitution modifier reduction largely as hp damage even though it's actually not. (This reader suspects this is an effort to speed play.)

When a creature sees its Constitution score reduced by 1 or more points and this loss subsequently reduces the creature's Constitution modifier, the creature suffers a reduction to its maximum hp (that is, the hp the creature has when it's fully healed). The DMG's absolutely right that will totally look like hp damage to a point. However, that point is really important because a reduction in a creature's Constitution modifier doesn't technically deal the creature any points of damage.

For example, Arboc is level 5 and possesses Con 14 and 25 hp maximum but has been dealt 20 points of damage so that he currently has 5 hp. He's dealt 2 points of Constitution damage, lowering his Con 14 to Con 10 therefore his Con modifier from +2 to +0. His maximum hp go from 25 to 15. Arboc, having already been dealt 20 points of damage, is now at -5 hp and dying.

On Ability Score Loss continues, saying

A hit point score can’t be reduced by Constitution damage or drain to less than 1 hit point per Hit Die. At 7th level, Mialee has 22 hit points when fully healed. Even if her Constitution score drops to 5 or lower, she will still have at least 7 hit points (less any damage she may take). (ibid.)

…And this is why tracking Constitution ability score loss that reduces a creature's Constitution modifier as hp damage is an unfortunate recommendation. See, a creature just can't typically have its maximum hp reduced below 1 point per Hit Dice or level. Even when a creature stright-up dies from having its Constitution score reduced to 0, that creature's maximum hp still equal its Hit Dice or level.

For example, Arboc has his Con reduced to 1 yet previously has been dealt no damage. His maximum and current hp plummet but only to 5 (equal to his level) and not, for example, to −24 (i.e. 25 his normal maximum −(7 his level × 7 for his reduced Con modifier)). Arboc could even continue adventuring normally despite his Constitution score of 1, although he'd likely tread very carefully and hide in the middle of the party.

To continue this example, Arboc is level 5 and possesses Con 14 and 25 hp maximum but has been dealt 20 points of damage so that he currently has 5 hp. He's dealt 13 points of Constitution damage, lowering his Con 14 to Con 1 therefore his Con modifier from +2 to -5. His maximum hp go from 25 to 5. Arboc, having already been dealt 20 points of damage, is now at -15 hp and dead.