Once you have broken the ice it's about building sexual desire. I've seen some videos about how to do that. I'm not familiar with this and I think it will take practice. I know that I should start with a gentle touch and good eye contact.

The latest weeks I've noticed that I have a hard time making enough eye contact (long enough, I tend to look away first) and I have rarely touched the girl because it's not natural for me to do that a lot. I've managed to win her trust but I haven't built sexual tension so far.

If you meet a girl every week by taking a walk together, what would be a good spot to touch her gently without making it look awkward?

2 weeks ago I found a good spot and I liked her reaction. We took a walk and suddenly someone nearly drove us off the road. She was frightened and 2 minutes later I took her shoulder and pulled her close to me, but just for a few seconds.
She leaned towards me and I liked it. But I let go pretty fast.

After that moment I haven't touched her and I have a hard time looking deep into her eyes. My eyes are the most attractive part of my body so I think I should give it a try. If it makes me smile out of nervosity, is that a bad thing?

eye contact is very important. it shows confidence and it really helps you read her mind.
somethings you could try is, before you see eachother come from behind her and cover her eyes and pull off a guess who... when she guesses your name (hopefully), then you take her by the hands and turn her around facing you, smile, small talk... go for a walk but hold hands.
if you don't want to hold hands then link arms. keep conversing, if she maybe starts to talk about something important, then you could just turn around face her and keep walking backwards while talking and hold eye contact.

it sounds like she is really comfortable around you. make your move quick before it may be to late!!!!!

after your walk with her, make a k-close!!