Admin said:

I think Nathan had a talent holding deal with ABC (meaning he got paid a set wage till they found him a job on their network) he'd done the guest appearances in Desperate Housewives Season 4 and someone showed him the script to Castle and he took one look at it and said THIS IS ME because he thought it was a perfect fit, I guess he did a read or something and they knew right away he was their guy… The story is out there somewhere but I'll have to find it.

Nathan told Andrew Marlowe & Rob Bowman "Stop looking. I'm this guy." Videos of the the story are at the 2010 ComicCon, and in the S.1 DVD set. It might even still be on the ABC Castle site.

He-Touches-Things and Admin are correct: Nathan read the script and knew right away he could do the part of Castle. 

Regarding Ryan's casting, Seamus said the following in an interview: "After they shot their pilot presentation, which was only 37 minutes long, they did some recasting and I think they wanted someone that would be John’s [sic] partner. They liked John [sic] but they didn’t have someone who was a match…was the right height…had a similar sensibility. Someone suggested "how about an Irish cop" and when I heard about it I jumped on it and did all I could to get in on it. That’s how it came about…They like to make fun of me some times that I came late to the party…'Oh, you weren’t there for the pilot, you missed such a great time.' "

Tamala also said that the part of Lanie was not originally written for an African-American woman: "My character "Lanie Parish" was actually written for a Caucasian actress, but my agent and manager along with myself were all in agreement that I needed to step out of the box a little bit. I auditioned for the role in the pilot and I booked it. "Lanie" was only supposed to be a guest star with a possible recurring role. The show got picked up by ABC for 13 episodes and by that time it was pilot season again. I was still auditioning for other pilots when I heard the news of the pick-up & was about to network test for another pilot. My team called ABC to let them know that I was about to test for another show & ABC put a stop to that, with an offer as a series regular role on what is now one of ABC's highest-rated shows.

All interviews are linked on the wiki.

I don't think the showrunners will name the actors and actresses who auditioned for the roles, but weren't hired until the show completes its run (not sure if that's a rule somewhere or industry standard, but it does seem to be common practice given previous interviews I've seen for other shows), but I'd love to know who was the other actress Stana beat out for Beckett and who else auditioned for the other parts. 

We don't know yet and I suspect we won't for a while.  The creators of Will and Grace wouldn't reveal who Debra Messing competed against for the part of Grace until the show was over (BTW, it was Nicolette Sheridan). 

I almost never watch the commentary episodes on the DVDs as it annoys me to have someone talking over the show, but I forced myself to listen to the one for the Pilot and that's when they tell the story of Nathan getting the part. They took the script to him in his trailer at Desperate Housewives and he told them to stop looking that he "was this guy". GREAT story.