Everything relates to everything else, so I don't believe this is a process that goes step 1, 2, 3. I think that you can start with any of these points and build from there.
My approach is dynamic, focusing more on answering questions about what the creature has to do in order to survive and thrive (which pushes their evolution), with their physical morphology and environment informing the answers.
All of these categories are based on evolution. Answer them and you have a start.
Mating, Pregnancy & Child Rearing. How does the species reproduce? What is the process used in order to get to mate and reproduce? Informing the answer will be what their evolutionary track is--bird evolution and eggs will produce a different cultural track than, say, a mammal-based live birth. This has to make sense biologically in order to track culturally. The answers to these questions will help get you to their psychology and societal structure, because it answers questions about general relationships in childhood and for romance, whatever that might mean. The physical characteristics of your creature, will help to inform this, from whether they are repilitain or birdlike or more like a mammal.
Questions to ask:
- How is mating achieved? What must they do in order to be considered
as a mate? Look to biologically similar creatures on this planet and
the rituals involved there. You can then build cultural custom out of
this. You can gather this info by researching similar animals on this planet.
- How long is pregnancy? What are the difficulties? Are any special resources needed to accommodate it? Again, look to animals as your model. Here I would widen research, looking at groups at large, such as mammal and reptile. Google is your friend.
- How many children are there?
- How long is the child cared for by the parents, if at all? Is extended family or the group involved, as it would be in a chimp troop?(This will
track into how developed they are at birth and some of their societal interaction).
- How are the responsibilities generally divided? Answer this based on
biology, as well as you can, finding an analogue in our own biosphere
as a starting point. (True, they can be alien, but we only have our
own biosphere to start with and the experiences of this planet. We
can reach beyond that, but if we are writing fiction it has to make
sense to the reader, who will also be from earth, as far as we know).
How much do they change over a lifetime? Do they look completely different when they are younger, like pupa and larva? Maybe spotted like a fawn until adulthood? These changes will be incorporated into culture--you might say a deerperson never really lost their spots to indicate immaturity. Those phases and the state of mind in those phases of development are going to be important to the underlying psychology.
What do they eat? Eating is a big part of the day and an important thing, so these questions determine societal cues you might not realize come from biology.
Omnivore? Carnivore? Herbivore? Insectivore? What do they eat and how
did they get it when they were more primitive--this informs behavior. Look at other creatures which share this taste for things.
If carnivore or omnivore,do they hunt cooperatively? Solo? Is there a division along gender
lines, as there is with the big cats? (females mainly hunt)
What physical characteristics do they have, which help them to get food, whatever food that may be? This can be anything from eyes that move independently of one another and a tongue which catches prey, to fast twitch musculature or a mouth shape built to efficiently eat grass. Ask too, when you're thinking about this, how these adaptations might effect how they interact with the world, each other and societal and cultural aspects.
What do they have to do in order to compete with each other for resources? Look into territorial behaviors or lack thereof for your species. This will effect psychology, ability to cooperate, and possibly their monetary system.
What eats them? In this category, this is anything that kills them, or did when they were primitive. This includes things that don't literally eat our creatures--like cold weather and disease.
What defenses does the creature have against the things that could
Same as above on physical characteristics, this time as applied to defense.
That reaction to those things will inform their evolution, and later, some of their behaviors, culturally.
The sea in which they swim This covers the environment in which they developed. Underwater, on the plains, or in the jungle--their environment determines how they look at the world, which, in turn determines their psychology, as well as their evolution.
Think about the environment. What's different? Underwater creatures will think more 3-D than plains creatures, and jungle creatures might not be very good at perspective (understanding that things in the distance are smaller, because the jungle is so dense, you don't ever get much perspective). The environment can color everything--how they hunt, mate, rear young, but that doesn't mean you should determine this first--it just means that if you come up with it last, you should make sure it fits with everything else you've done. Look at all the animals you can in that environment.
Think about the adaptations needed in body, and how that affects their interaction with the world.
Lifespan. Knowing how old your species gets and if they get infirm is critical to societal structure. If, like salmon, they tend to die post spawn, this will effect everything, as will species which tend to have a place in their society for the aged (chimps often do, but it is dependant on gender).
How old do they get?
What happens when they get past breeding age, biologically?
Do they have a role to fulfill when they are older?
Are there any periods of time when the species is especially vulnerable? For humans, we have a long childhood, but for an alien species, this can be a chrysalis, or shedding skin or anything really. It can happen because they are old, or any number of biologic reasons. Ask what the species does/did during this time, and how that effects their way of looking at the world, as well as how others in the species react to it.
Are they dependant on/do they need any one thing/creature? Do they have a relationship with any other species biologically? Even a crocodile has birds that clean his teeth. Are they the bird? Or the crocodile? Or neither. This relationship can inform their psychology...Is there something specific they need in order to survive? Most specialists die out, but kolas need eucalyptus--it's all they eat...your creature might have to eat stones for better digestion, or eat a particular type of plant periodically. Horses, for example, do need salt, so they end up licking each other sometimes to fill that need. Look at the need/dependance through a social and psyche lens, once you understand what it is. It can have a huge cultural impact on custom.
When will you be done? Well, you might not be. There's always something new you can discover about your creature. Gathering the info is just a matter of google, the library, and answering questions. Each of these questions and steps will be informed by the others, like a giant damn circle. Just pick a place to start and come back around. I know you were hoping for something more step-by-step, but a creature creation is part research and part creative process.