My character has to perform surgery on herself and by herself.

She has all the surgery tools and a screen to watch what she is doing.

She is in a clean stable environment and there is no risk of infection.

She has to perform surgery near her spine, where she's inserting a microchip connecting to the nerve connected to the brain. The technology is futuristic so this is possible.

What I'm specifically asking if she could actually do the surgery?

Yes, this is possible.

Here are some real life examples:

Woman carries out caesarean on herself - and she and baby live
https://www.theguardian.com/world/2004/apr/07/health.healthandwellbeing

A pregnant woman in Mexico is reported to have given birth to a
healthy baby boy after performing a caesarean section on herself with
a kitchen knife...The unidentified 40-year-old [other sources give her name as Ines Ramirez Perez], who lives in a rural
area without electricity, running water or sanitation, and whose home
was an eight-hour drive from the nearest hospital, performed the
operation when she found she could not deliver the baby naturally.

She took three small glasses of hard liquor and, using a kitchen
knife, sliced her abdomen in three attempts ... and delivered a male
infant who breathed immediately and cried...Before losing
consciousness the woman told one of her children to call a local nurse
for help. After the nurse stitched the wound with a sewing needle and
cotton thread, the mother and baby were transferred and treated...at
the nearest hospital."

Doctor rescued from Antarctica in 1999 dies at 57
http://www.cnn.com/2009/HEALTH/06/23/obit.jerri.nielsen/index.html

Dr. Jerri Nielsen...treated herself for breast cancer while stationed
at the South Pole in 1999...Nielsen caught the nation's attention in
1999, when she found a lump in her breast as a 47-year-old physician
stationed at the Amundsen-Scott South Pole Research Station. After
finding the lump in June, she diagnosed herself with breast cancer and
began treating herself using chemotherapy agents that the U.S. Air
Force parachuted to the station the next month.

It was later revealed...that Nielsen -- an emergency room doctor from
Cleveland, Ohio -- performed a biopsy on herself with the help of
non-medical crew, who practiced using needles on a raw chicken.

In your character's case, the surgery is something from the future so hopefully the tools are too. She'd be unlikely to be able to reach her spine properly but even today we have robots that do surgery and can be controlled by computer. So she could do that.

Leonid Rogozov, when based on a Russian Antarctic base as doctor, was forced to perform appendix operation on himself.

On the morning of 29 April 1961, Rogozov experienced general weakness, nausea, and moderate fever, and later pain in the lower right portion of the abdomen. None of the possible conservative treatment measures helped. By 30 April signs of localised peritonitis became apparent, and his condition worsened considerably by the evening. Mirny, the nearest Soviet research station, was more than 1,600 km (1,000 mi) from Novolazarevskaya. Antarctic research stations of other countries did not have an aircraft. Severe blizzard conditions prevented aircraft landing in any case. Rogozov had no option but to perform the operation on himself.

The operation started at 02:00 local time on 1 May with the help of a driver and meteorologist, who were providing instruments and holding a mirror to observe areas not directly visible, while Rogozov was in a semi-reclining position, half-turned to his left side. A solution of 0.5% novocaine was used for local anaesthesia of the abdominal wall. Rogozov made a 10–12 cm incision of the abdominal wall, and while opening the peritoneum he accidentally injured the cecum and had to suture it. Then he proceeded to expose the appendix. According to his report, the appendix was found to have a dark stain at its base, and Rogozov estimated it would have burst within a day. The appendix was resected and antibiotics were applied directly into the peritoneal cavity. General weakness and nausea developed about 30–40 minutes after the start of the operation so that short pauses for rest were repeatedly needed after that. By about 04:00 the operation was complete.

After the operation gradual improvement occurred in the signs of peritonitis and in the general condition of Rogozov. Body temperature returned to normal after five days, and the stitches were removed seven days after the operation. He resumed his regular duties in about two weeks.

But in this case the abdomen is accessible with the hands with no big hurdles. Accessing the back region is going to be more cumbersome, if the operation has to be executed by hands. If instead your character has the necessary tools for performing tele-surgery (basically remotely controlled robotic arms and hands), it should be feasible.

It sounds like the most difficult part would be actually reaching around to to insert the microchip if it's behind her own back. With a local anaesthetic, she'd be able to make the incision and insert the chip. Obviously I can't speak to the extent of the futuristic technology, or how involved the surgery is, but a simple insertion certainly wouldn't be too hard.

You might be interested in reading about a Russian doctor called Leonid Rogozov. He was the lone physician on an Antarctic expedition when he developed appendicitis. With no other options he gave himself a local anaesthetic and performed a self-appendectomy.

In his case he did have assistance in the form of a driver and meteorologist, but he made a full recovery and resumed his duties about a week later.

Another related topic would be biohackers or grinders, which is a term for people who experiment with cybernetic implants upon themselves. An example would be inserting small silicon-coated magnets under the fingertips, which can give the person a primitive electromagnetic sense as the magnet twitches and pushes against the nerve endings in the presence of an electrical field.