As Lees says it is a difficult job although not impossible, but the result is not worth the effort. You could get far better results with an aftermarket install and it will be easier.
All the speakers need to be changed, which is the easy bit, although you will need adapters for the rears as they are only 4" drivers.
The tricky bit is wiring in the amp. Your current system has the front speakers running off the HU and the rears and sub running off the tiny amp inside the sub.
The Bose system has all cabin speakers running off a 4 channel amp mounted in the offside cubbyhole in the boot, with the sub having its own amp inside the tupperware enclosure.
The wiring is completely different, so unless you have the full wiring loom with the system (unlikely) you will have to make up your own loom.
This will involve using the RCA adapter from the HU, running 2 sets of RCA leads to the boot, running a higher power cable to the boot for the amp and sub. You'll then have to solder all the new cables onto the amp connector, with a third set of RCA's (+power cable and remote on lead)ruunning to the Bose sub.
To fit the sub you need to strip out the rear speaker panel, boot lip panel and LHS boot trim panel.
You then need VAGCOM (or trip to Audi dealer) to reprogram your HU for the Bose settings.
With all that done the results will be disappointing.
A much better upgrade would be to replace cabin speakers with decent aftermarket units (there are plenty of options). Then install one of my 300w Stealth subs with a new Alpine amp driving the new sub and rear speakers and hidden behind the rear speaker panel. This will provide clear punchy bass, allow the HU bass setting to run at -2, which means you can drive the cabin speakers harder with no distortion, while keeping the OEM look.
This upgrade would cost about £400 including installation & reasonable Kenwood or Alpine cabin speakers and you will then have a decent system far better than the Bose upgrade which will involve less work. I have fitted this upgrade to a number of S3's with or without Bose, and the results are amazing. It also provides a good platform should you decide to get an aftermarket HU.