I've had better results on Black Walnuts by planting the nuts over using bare rootstock as supplied by our state conservation department. You avoid transplant shock and all the losses associated with that method.
First thing to do is to decide on what nuts to use. Since you stated that you have some trees on site, if you have been able to observe them for a number of years you have probably noticed that some consistently bear a better crop than others. The one with the best traits would be the one I would pick to be the mother tree. Collect said nuts when they fall and save for planting later the same season.
You could prepare for late fall planting ahead of time some where during late summer. Each individual planting site (2'x2' or 3'x3' for example) could be sprayed with Roundup to help get rid of future plant competition. Another application of Roundup could be applied in spring before saplings emerge to get rid of any vegetation that came out during winter or early spring.
I would plant in late fall, the way I do it is to use a pick bar at each site poke a couple of holes and push the 2 nuts in 2" to 3" deep, tamp a little dirt back around the holes. If both sprout I let them both grow until until I can see if one is stronger and then cut the weaker off.
Spacing will depend on what you are trying to achieve, nut or timber production. With closer spacing you can always thin later while in the mean time getting some nut production and trying to establish a dominant canopy(walnut) to reduce undesirable woody competition.
Since I am no typist and I've pecked out about all I can stand to for now I think I'll stop here. If this helps any and you have any specific questions just ask and I will do my best to respond.
Best of luck Swannie