This is fairly straightforward. Unscrew the screws along the bottom, 3 screws on the edge (nearside to steering wheel). Then unscrew the bolts using an Alan key  (3 inside inner door handle). Then 2 bigger ones by trim panel, there's also a screw in the middle of the trim panel that needs removing.

 Once all the screws/alan key bolts are undone, remove the electric window switch by prising it out with a screwdriver or another flat bladed tool. Unplug the two switches for the left and right windows.

 Now you should be able to pull out the plastic trim panel and reveal something looking similar to the pic on the right. You'll have to partly remove the sealant wrapper to expose the right side of the door (use a sharp knife to cut through sealant adhesive as needed). May have to break some of the sealant sheet if clips won't come out.

 Undo the screw as marked, you'll need a 6 point star screwdriver (why the fuck Fiat can't just use a normal screw I don't know). You'll also find an aperture in the door frame to un a bolt holding the handle on (marked area in first picture).

Once the bolt and the screw are undone you'll be able to wriggle the door handle out (believe me,  it does go). There are 2 long retainer pins attached to the door handle, you'll need to detach these (best way to break the plastic holding clip). 

 You may receive 2 long pins (as marked in pic) with the lock barrel, you probably will not need these as there are already 2 in the door (however you will need 2 plastic retainer clips as they'll probably break when you take the old handle off). The 2 long pins are used to operate the door locking and opening.

To remove the lock barrel from the door, use a small nail and a hammer to remove the hollow Pin A from the door handle (its right by lock barrel). Once you've done this you should be able to undo a horse shoe spring as shown in pic (object B).

 Put the lock barrel into the new handle and add the retainer spring (B).  Carefully prise the door handle back in, you should be able to connect at least one of the long pins  before putting the handle back in properly.

Getting the 2nd pin into the plastic holding clip can be a bugger, if you're lucky you may be able to put both pins in place before putting the handle in - I had to put the first in, then from the inside squeeze my hand through a narrow gap in the door and manually push it in.

 Tighten nuts, check it works and the put everything back. If need be re-add sealant to plastic covering sheet (I used mazdik)