My favorite specific example of alien psychology comes from Peter Watt's first contact novel Blindsight
available for free on his website. The story is good and the copious amounts of exposition never feel slow (the characters' stated motivation is to learn about the aliens, so every "infodump" is less an interruption from the plot and more of a development in the plot itself), and for those readers who love hard SciFi exposition for it's own sake, the
Notes and Acknowledgements at the end is as hilarious is the story itself. If you're the kind of reader who enjoys that, I would recommend reading it first. I did. Actually, a lot of it was beyond my level, but I skimmed the whole thing. The same man also wrote a John Carpenter fanfiction
The Things that does a fantastic job of showing just how
wrong we look to the alien, and these two stories were the main inspiration that got me thinking about non-human psychology in my own SciFi/Fantasy.
Although the main
method that I use comes from a TVTropes article
So You Want To: Design An Alien Mind and a
Giant In The Playground post that references the same article: Rearranging Maslow's Hierarchy. Humans care about Physiological Needs first out of five, then Security Needs second out of five, then Relationship Needs third out of five... but maybe another species needs Relationships first and Physiological Needs last. We wouldn't understand why that species doesn't fear death and give it their all to live longer, but they wouldn't understand why we
do care about making ourselves miserable delaying the inevitable instead of making the best of the time we have.
This clearly isn't a common method – I found the TVTropes article years ago, have been looking for "rearrange Maslow's Hierarchy science fiction" ever sinve to see if other people do it, and to this day, the GiantITP post is the only one I've found besides my own postings – but it's a method that I've had great success with.