When to go
Best season
September, October, November
To capture the yellow leaves (like I have in my photo), I recommend visiting Burg Eltz from the Cross Chapel viewpoint in the fall.
In my opinion, you should definitely wake up early and make your way to the castle at sunrise. It can get very crowded during the day, so going as early as possible is the best way to avoid tourists. You'll also have the best chance of capturing some of the morning fog rolling over the hills at this time.
Field tips
Make use of the natural formation of the leaves to guide the framing of your photo. Not only will the leaves add some depth and dimension, they'll also bring a pop of color to the photo.
Try to use the pathway to the entrance of the castle as a leading line.
You'll be able to capture the forest and the entire castle with a wider lens (e.g. 34mm).
After getting this perspective, head down to the castle's entrance and take some photos there. I have another locations here in the app, so check out it and see the other tips I have for shooting at this location.