How can I mount my iPhone 6 in the cockpit to record my flights?
UPDATE  9/4/2017
I'm thrilled this post has been so helpful for so many people. As such, I wanted to  update it with more details and more current links in case people are interested. Enjoy!

You could have a lot of fun, and get a lot out of reviewing the footage of your lessons. I'd look into a suction cup mount as you are already considering. When buying a suction cup mount, make sure you get one that is &quot;beefier&quot; than you expect you need. Whatever you do, make sure you share your videos online, your video might even get someone else interested in aviation!
Below are the details of doing what you asked. You might be able to buy a used, older model gopro for a comparable price - but I'd actually argue your phone is probably a better option.
What to buy:
(this all actually gets kind of expensive, but is still cheaper than buying a go pro and accessories)
Must have:
To get GoPro functionality from your iphone, you'd need a mount and a case:

http://amzn.to/1ZQDx7q (Optrix Super Sucker Mount)
http://amzn.to/1MyN1uc (Case)


This is a mount that mimics the functionality of a go-pro mount. It also let's you know if pressure is dropping in the suction cup as you climb.
If you ever feel brave enough you could even stick this to your wing!

http://amzn.to/1GmeYbU

The view angle probably is not wide enough. This inexpensive kit will apply a wide angle to your iphone. I'd make sure your case/mount doesn't preclude using one of these.
Best alternative camera to GoPro:

http://amzn.to/2vZsQEx (Garmin Ultra Virb 30)

I love the aviation data overlay of this camera. It works very well and helps debriefing a lot. I personally prefer this over my GoPro because it includes that data.
Best GoPro options:

http://amzn.to/2xJKn67 (GoPro Hero5 Black)

4k resolution @ 30FPS, 12MP quality is great. I use this mainly for wing mounted shots.

http://amzn.to/2eDiQi3 (GoPro Hero5 Session)

The nice thing about the session series is they are small and super simple to operate. I use this camera to point inside the cabin, on the side windows. It is nice because it doesn't block as much of my view outside, even in a mount on suction cups.
Important Accessories:

http://amzn.to/1ZQERqT (Zoom H1)

I use this to record ATC/Headset audio. It is easy to have it recorded to its own device and lets me have better control over the audio recordings than if it was part of a video file. The audio is oftentimes more useful than the video alone, especially with your instructor explaining things to you. Having a separate device also reduces your point of failure risk if you have a camera battery die.

http://amzn.to/2wAg29z (Headset cable to record audio)

You need this cable to &quot;grab&quot; audio from your headset jack to record it to a camera or external recorder. (Note- the Garmin Virb 30 Aviation bundle includes this)

http://amzn.to/2wAo1Uh (External Battery)

You will be running out of battery fast if you record HD video for any length of time. This external charger will keep you from running out of battery.

ND (Prop) Filters (remove the weird prop strobing from your footage)

(Note- the Garmin Virb 30 Aviation bundle includes this)

http://amzn.to/2xJhxm9 (GoPro Hero5 Black ND Filter)
http://amzn.to/2exTIFG (GoPro Hero5 Session ND Filter)

Placement:
The best place to put it would be between the seats on the ceiling pointing landscape style towards the instrument panel. This would give you a view of all of your instruments and maybe even a bit of the horizon. At first, we all struggle to use the horizon as a reference instead of instruments, so putting the camera close to head level might help develop a good sight picture.

Here's a video using the angle I am referring to.
Unless you set up a second camera, you are unlikely to get rudder input on film. This video shows a good example of that angle, but might be impossible while getting dual instruction.

Video showing rudder control input on takeoff
Words of caution:

Make sure your filming doesn't distract you. This is really serious and not something to shrug off. Remember, you don't want the FAA to have footage to shove in your face of you being reckless.
Make sure you don't damage the aircraft with the mount. The windshields in particular are vulnerable to being squeezed out of shape from the pressure in the suction cup, especially if they are hot. I have seen some indented windshields recently due to gopro mounts, which makes it very hard to get a normal sight picture (imagine looking through a fun house mirror).
UPDATE
Another issue I recently saw was a result of someone using this camera: http://amzn.to/1oos68G. The camera has a built in rare earth magnet which made mounting it very easy. The problem was, it was impacting the magnetic compass - so be aware if you get a magnetic mount to make sure your compass is not affected by it being placed in the cockpit! When attached to the overhead speaker we saw no visible impact on the compass, but worth keeping an eye on.

Take a look on youtube at some popular pilot channels to see how they record. From there, you can pick your favorite angles and splice a film together.