Often what writers do when they try and make aliens really alien is to make them incomprehensible, they act is ways that don’t make any sense to the characters. This will often fall flat, there are certain things that even aliens must have if they have a technological civilization (Note that if a race was uplifted these need not apply).
First they must have a survival instinct, if they give up on life too easily they would hardly still exist.
They also must have a sense of community, a need for social contact. You can’t build a civilization working by yourself. They would almost certainly have to raise their kids.
Third and most important they must be curious, if they are not they would never do any science, they might never even get to the fire and wheels stage.
They also have to be both intelligent and rational enough to do the science and engineering needed to get out into space. We may meet aliens that we can’t have a talk about love with, but I doubt we will meet aliens we can’t talk about math with.
The thing is though that your aliens need to be different, but still have the above and still able to interact with humans to some degree (or they would probably not exist in your story).
For my money there are really only two things you need to decide; goals and motives. Everything else will take care of itself.
First give your aliens some goal, something that we can understand, but may not know, something we would not do ourselves. For example the Borg in Star Trek have a simple goal “Multiply” that is really it, everything they do supports that one goal.
Their motive is “To achieve perfection” not that they define perfection like we do, but they don’t have to think like we do.
One of the reasons the borg are so effective as villains is that you know why are doing what they are and that they will never stop. They are a force of nature that you can’t even hope to reason with.
There are any number of motives and goals you could give your aliens but the most important thing is that YOU understand them. You don’t have to tell the reader but if you don’t understand them then they will probably not make any sense to anyone. As a rule simple is best, it should be something that can be expressed in one sentence.
One way to get across this difference when dealing with them on screen is the way they talk. Don’t have them talk like a human, have it clear that whatever is translating is having to work hard. The best example of this I can think of is the Vorlons from Babylon 5. When they speak you hear a set of almost musical notes then something like “The avalanche has already started, it is too late for the pebbles to vote”.
You don’t have to go that far but it should be clear that the person talking is not human even if they are using a translator.