The fence isn't the concern as much as the house(s).

First off, what kind of fence is it? Will the neighbor let you take it
down?

Second, why so deep? Why not 4-5 feet?

How far away are the houses? Yours must be within 6 or 7 feet or where
you will be working. Do you have a basement? How deep are your
footers?

Can you go around the other side of your house to keep away from your
neighbor?

What kind of soil do you have?

Where will the water go? Is it legal? Do you care?

That said, where I live, we have lots of water. We have about 2 feet
of crappy soil over pure clay. So for me, there would be no use in
going down over 2 feet -- just to hit the clay line. Going down 10
feet seems awfully deep.

At 10 feet, you have to worry about collapses. If the trench goes,
will it effect your house (or your neighbors)?

Do you have a septic tank? If so, don't send water towards your
leachfield.

Also, if you start on the bottom and work uphill, it might help contol
your water during excavation. And don't forget some hay bales or
plastic to control erosion and sediment.

In all likelihood, you will need a backhoe with an extendable arm so
you can reach past your A/C and not have to be right next to it.

Good luck. If you give some more info, someone might be able to help
more.