New Transform Rules and Archangel Avacyn
No, Avacyn will not transform again at the following upkeep.

The new rule 701.25e says


  If an activated ability of a permanent tries to transform that permanent, the permanent transforms only if it hasn't transformed since the ability was put onto the stack. The same is true for triggered abilities of a permanent that aren't delayed triggered abilities. If a delayed triggered ability of a permanent tries to transform that permanent, the permanent transforms only if it hasn't transformed since that delayed triggered ability was created.


Avacyn's ability creates a delayed triggered ability, so the last clause is the relevant one. So, the situation you describe plays out like this:


A non-Angel creature dies. Archangel Avacyn's triggered ability goes on the stack.
Archangel Avacyn's triggered ability resolves, and creates a delayed triggered ability.
At the beginning of the next upkeep, the delayed triggered ability goes on the stack. 
Another non-Angel creature dies. Archangel Avacyn's triggered ability goes on the stack.
Archangel Avacyn's triggered ability resolves, and creates a second delayed triggered ability.
The original delayed triggered ability resolves. Archangel Avacyn has not transformed since that delayed triggered ability was created, so Archangel Avacyn transforms into Avacyn, the Purifier.
At the beginning of the following upkeep, the second delayed triggered ability goes on the stack.
The second delayed trigger resolves. Avacyn has transformed since the second delayed triggered ability was created, so she does not transform, and remains Avacyn, the Purifier.