The NUMBER ONE rule for sewing with french seams is to make sure you are using a somewhat thin fabric. I've attempted them before on fabrics like bottom weight denim and the outcome is bulky and happy-hands-at-home looking.
1. Sew a line of gathering stitches onto the prepared sleeve. I actually like to sew 2 lines with a 4mm stitch length and leave the thread ends long enough to pull.
2. Pull one set of threads to gather the sleeve slightly. You can always gather it more when you get it situated in the sleeve but this gives you a little head start.
3. When I'm sewing french seams that will intersect, I always iron them in different directions. For example, here where the side seam meets the sleeve seam, I've ironed the side seam towards the back and the sleeve seam towards the front. Even in an especially thin fabric, french seams cause some bulk, and ironing them in different directions cuts down on this.
4. Pin the sleeve to the bodice WRONG SIDES TOGETHER. Line up your notches and adjust your gathers. I use long pins so I double insert them for a good grip. Don't cut your long gathering threads yet. The gathers on this sturdy cotton aren't going anywhere but they could slip out in something silky.
6. Stitch. When you get to the area with the gather, try to smooth out any bumps to avoid puckers.
7. When you get done with the first seam, inspect your work. Any big tucks need to be undone, readjusted and restitched.
8. Press. I pull the sleeve out so that I'm ironing on the front bodice and pressing the seam towards the sleeve. Pull on the sleeve a little to ensure a crisp edge for the second seam.
9. Here's what the first seam should look like when you're done pressing.
10. Trim away about half of this first seam allowance. I always put my hand under the edges being snipped to that I don't catch my bodice fabric on accident.
11. My sleeve is done being trimmed.
12. Turn the garment to the inside and pin for the second seam. Adjust gathering threads if needed. I like to sew this seam with the gathers up so that I can adjust any areas that might need it.
13. Stitch, keeping the needle in the same position as the first seam and aligning the fabric with the right edge.
14. The finished sleeve seam. Inspect for tucks and clip the gathering threads off.
15. Pull the sleeve through the armhole and iron from the inside, pressing the seam towards the sleeve.