In Koine Greek, how are verbs conjugated based on their tense (if there is any pattern at all)?

Not quite clear what your question is. Assuming I’ve understood it:

Koine Greek, like other languages, has a notion of principal parts. There are six tenses you need to know for a verb; once you know them, you can derive the remainder. The six tenses are all indicatives: present; future; aorist active; perfect active; aorist passive; perfect passive.

There are in fact regular classes of verb, derived from the verb root; but there’s a lot of morphophonology happening at the interface of the root and the tense suffix, so you need to be familiar with what the possible tense stems are.

To grapple with the possibilities, read this: How can I learn to individuate ancient Greek verbs? It’s written for Attic, but Koine is not substantially simpler than Attic.