In Godfather 3. She was surely not an outsider, because she also participated in ordering the death of Zazaa and she asked Vincent to 'strike back'. She must be part of the family business. But what was Connie's definite position in the family business?

We are not told what, if any, position Connie had in the family business. Probably she had no official position. After her reconciliation with Michael in the boathouse during Mama's wake in GFII, I inferred that she became a general busybody and caregiver.
One of the noteworthy developments of GFIII was that both Connie and Kay became vastly more assertive than they had been. Others on this board have attributed that development as an exemplar of the growing assertiveness of women. They're probably right. But I doubt it it had any effect on Michael, the hyper-controller. No way would he give Connie an active role in his business. So she asserted herself in other ways, such as pushing Vincent on Michael.
Another of those ways was to conspire with Vincent and Neri to whack Zasa while Michael was in a diabetic coma. This wasn't a strategic business move, or an example of Connie stepping in and taking the reins of the business while Michael was incapacitated. It was probably personal on all sides: Connie (revenge for the Atlantic City attack, "kill-or-be-killed" for her ailing brother); Vincent (seeing an opportunity to get rid of his own mortal enemy and rise in the family); and Neri (better go with Connie and Vincent in case Michael dies).

There is nothing to indicate any definite position that Connie held in the family business. Since the Corleones were a "family" within a family, anyone in it who wanted to assert him or herself probably could do so. Connie did so at some point. remember, she is Vito's daughter. She got at least some of those genes.

By the time GFIII rolls around, Connie has become what seems to be the "Matriarch" of the Corleone family. As both TB and Olivant have said, there is nothing that indicates that she held a "titled" position in the Corleone family business.

No question that Connie's role has become much stronger by GFIII. Although she holds no title, she is by far the most owerful woman in the history of the Corleone family. She engineered the succession of Vincent to becoming the Don, and IMHO she helped move Michael toward stepping down. Once he told her about having gone to confession Connie immediately refers to Fredo's death, and we get the sense she realizes that Michael no longer has the strength to lead the family. When Michael cedes the Donship to Vincent, Connie enters the room and VIncent looks at her and simply says "It is done." So clearly the two of them conspired in some way to make all of this happen. Note also that she seems unhappy that Michael exacts a price from VIncent--- namely that he gives up Mary --- but Connie is too smart to interfere with this.

Finally Connie is responsible for the Canelloni "hit" on Altobello. I believe this is the first and only arranged hit by a Corleone woman (with the exception of the failed attempt by Vito's mother).

interesting comments ALL. This brings another question to mind... In the world of "La Cosa Nostra", is this type of thing even permissable???

Good question Frank.

I don't think that back when GFIII takes place, a woman would be allowed to interfere with the "family business." However the Corleones were not the most orthadox mafia family.

As far as more recently, in real life, here are two stories that may answer your question:

Mob Woman's Shootout Leaves 3 Dead

Italian Mafia Sees Rise In Girl Power

Wow this opens up a whole new genre..."The Godmother"

Thanks for the insight DC!

I don't believe Connie and Vincent had even thought of pulling Micheal down. She referred to Fredo because she just wanted to appease Micheal. And I guess 'It's done' means 'The order to kill has been given' , never 'I have replaced Micheal'. Connie always loved Micheal and Vincet respected him, as can been seen in the film

The "it's done" line means that Michael has finally accepted Vincent as a Corleone and has handed the reigns over to him.