The exact word you are asking about is κρίνω, which is actually rarely used in the New Testament, only three times outside the gospels. However many other derivative of the same word translated as ‘judge’, ‘judges’, ‘judgment’ that all carry the same root meaning. I will therefore answer the broader root meaning of them all.
According to Kittel’s TDNT the basic sense in Greek literature is ‘to part,’ ‘to sift,’ which leads to the sense ‘to divide out’, ‘to select’, ‘to value’, with the most common meaning is ‘to decide’. From this root is also means ‘to judge’, ‘to assess’, ‘to go to law, to dispute with’. Also ‘to seek justice,’ or ‘to be accused’.
In the LXX (Greek Old Testament) it is predominantly used for ‘legal’ words. It is used for the Hebrew שׁפט which carries the double sense ‘to rule’ and ‘judge’.
From these roots we can understand that the word, when used is basically to asses and resolve with the implication of being being able to be taken under the idea to ‘rule’. Therefore that basic meaning that can transition into the context of both official judgement and personal judgments carrying a legal tone. Here ‘seeking justice’ can easily become a personal door to bitterness, arrogance, assuming a higher position of ourselves over others, etc.
One verse outside of the gospels that used the exact word in question shows the usage of all these sense all together:
But with me it is a very small thing that I should be judged by you or by any human court. In fact, I do not even judge myself. For I am not aware of anything against myself, but I am not thereby acquitted. It is the Lord who judges me. Therefore do not pronounce judgment (κρίνω ) before the time, before the Lord comes, who will bring to light the things now hidden in darkness and will disclose the purposes of the heart. Then each one will receive his commendation from God. (1 Co 4:3–5, ESV).