Ndtyhij

Ashildr pilot a TARDIS

with Clara, in the end of Hell Bent.

But, except for short or emergency flights, normally with the presence of the Doctor alongside, no human (River Song isn't pure human) would have been able to pilot the Doctor's TARDIS by themselves. There are many difficulties with the way it's portrayed:

1. Ashildr is shown reading the manual, which is written in Gallifreyan, arguably one of the most complex and difficult languages to learn and understand in the universe. I'd imagine a cross of Chinese with Category Theory (in math) might be similar to nursery level in Gallifreyan. Not only the language is difficult, many of the concepts probably don't even exist in human language.




2. Piloting the TARDIS (safely, at least) should depend on an extensive knowledge of temporal mechanics which would have been beyond the capabilities of a human brain, no matter how long they'd have to study it. The Doctor was seen trying to teach Clara how to pilot the TARDIS, yes, but the "safely" is still open for discussion. The TARDIS can destroy the Universe in case something wrong happens (see Big Bang).




3. Being human, both lack the Time Lord temporal senses, which may have something to do with piloting the TARDIS as well. Like having eyes to be able to fly a plane.




4. Most crucially, I think, as non-Time Lord they'd lack the symbiotic nuclei, which AFAIK are necessary to establish the symbiotic bond between the TARDIS and its pilot (I believe it's said in PROSE: Interference - Book One).

The TARDIS will die if its pilot dies, so it's not like someone can simply take over a TARDIS. They need their pilot. River Song has been modified by the Silence with many Time Lord characteristics, including the symbiotic nuclei which are also necessary for the regeneration, besides having a connection to the Vortex from the moment of her conception. Even so, she needed the Doctor's TARDIS teaching her. How did Ashildr and Clara manage it, with a bog-standard TARDIS which wouldn't be accostumed with non-Time Lord passengers, let alone a non-Time Lord pilot?

Another point is that with all the trouble the Doctor finds traveling in the TARDIS, having a pair of humans flying around with a TARDIS of their own is a serious risk of universe-destroying paradox. For example if Clara manages to get "killed" along the way, with no Doctor to save them. Remember the Doctor deactivated Jack's Vortex manipulator for this reason, and Jack was a trained, and experienced, Time Agent.

Ashildr is not flying the TARDIS. Clara is.

From the very end of the episode (emphasis mine)

CLARA: Mind you, seeing as I'm not actually ageing, there's a tiny little bit of wiggle room, isn't there?

ASHILDR: Wiggle room?

CLARA: Wiggle room. Yeah, you know, wiggle room. We could, er, you know, stop off on the way.

(Clara sets coordinates.)

ASHILDR: Where are we going?

CLARA: Gallifrey. Like I said, Gallifrey. The long way round.

Clara sets the coordinates, and pulls the lever to send the TARDIS flying. Sure, Ashildr is shown reading the manual, in Gallifreyan, but only to try to fix the Chameleon circuit (which seems to be a standard failure in Type-40 TARDISes). But at the end of the day (or the episode), Clara is flying it.

You might ask, How? Well, who knows...after several years watching the Doctor do it, she might have picked up something. That's how I learned to drive with a stick...by watching first.

EDIT:
After over-analyzing it, I can think of a couple instances when Clara "drove" the Doctor's TARDIS:

1) The TARDIS Telepathic Interface: In series 8 episodes "Listen" and "Dark Water", the Doctor "plugs" Clara into this interface, and she effectively "flies" the machine. The Doctor states so himself:

The Doctor: Don't think about anything. Remember, you're flying a time machine.

2) At the end of series 8 "Kill The Moon", after Courtney Woods leaves, the Doctor is ready to take off again, but Clara pulls a lever and the TARDIS stops. At least she knows where the brakes are.

3) In the following episode ("Mummy in the Orient Express), at the very end Clara is seen helping the Doctor to take off.

Clara was seen using pretty much all of the Doctor's toys, or gadgets during her run as a companion. She learned how to use both the Sonic Screwdriver and the Sonic Sunglasses, and even the psychic paper. She handled herself in UNIT HQ like she owned the place, even giving orders to Kate Stewart.

So as I see it, there are two options:

a) Clara is using the Telepathic Interface in her new TARDIS

Or

b) She learned how to fly it by watching the Doctor.

Personally, I like the first option better.

Clara can pilot the TARDIS with Ashildr because:

1. The Doctor taught her some basic commands (from Journey to the Centre of the TARDIS onwards, at least), enough for travelling from time/place to time/place, though most likely not enough to access full TARDIS functionality;




2. This TARDIS is a new, tame one, unlike the Doctor's TARDIS whose rebellious nature is probably part of the reason she had been retired by the Time Lords. It's a newer model as well, since the Type 40s were already old when the Doctor flew off Gallifrey for the first time (he says his was a museum piece). It would simply take them to exactly the coordinates they ask for, if it is safe enough;




3. The TARDIS can be programmed with automated flight parameters, like an autopilot, instead of just "manual" piloting. This would be what they're using most, like when Clara set the coordinates in the console. It may also have some sort of "Instrument Flight" mode, making up for their lack of temporal senses;




4. The unmodified TARDIS would have enough basic safeguards for normal time-travel without complications;




5. The new TARDIS was flown first by the Doctor, so he probably activated it himself, setting himself as its "owner". It might still malfunction without him or another Time Lord around, but it may take some time;




6. It's possible the Time Lords neglected to allow for the possibility of passengers stealing a TARDIS, when designing the safeguards. Or the Doctor disabled them.

For examples of Clara flying the Doctor's TARDIS (alongside him), please see tilley31's answer. Reading Gallifreyan continues to be a problem, but there are several instances of it happening somehow, and they wouldn't need to understand everything anyway.

So, as long as they don't act recklessly, it would be safer than flying the Doctor's TARDIS. It won't protect them from the Time Lords, though, if/when they decide to track their TARDIS through time and space.

ANYONE can pilot a TARDIS once it's been primed with the Rassilon Imprimatur (see The Two Doctors) It can be assumed that the TARDIS they are using, having been stolen by The Doctor a brief time before, was a working model that did not need the Imprimatur.

As for how she's flying it, she read the manual. She was going over the portion of the manual about the chameleon circuit when Clara returns to the control room.

The TARDIS is certainly translating the text of the manual for her, as it would for any Companion.

The symbiotic nuclei is required to pilot a TARDIS as was demonstrated in "The Two Doctors," but this rule was broken before. Like 12 says when he steals the chair from Davros and enters the room full of daleks with a cup of tea, "I'm the Doctor. Just accept it." Pretty much every "rule" on the show gets broken. Simply enjoy it.

The Doctor is a "super intelligent alien being who flies through time and space," but he is unsure if a cold star can exist in "Amy's Choice." With all that intelligence, he must know the first law of thermodynamics. The coldest temperature possible is absolute zero. Space is already absolute zero. A kid in a highschool science class would know a cold star can't exist. Yet, to make an interesting story, 11 conveniently forgets this law.

I could go on and on listing examples of inconsistencies in Doctor Who, but I won't. I love the show and have seen all 37 seasons too many times to count. Yes Clara piloted the TARDIS without a timelord present. Oh well. In "The Rings of Akhaten," 11 says it best - "it's a nice story."