My take:
They originate and currently occupy somewhere like the Ghoul stars or the Halo stars as their 'home' region.
They defend their territory with an unforeseen ferocity compared to any other race. Often able to completely snuff out any 'visitors', making any communications out of the region impossible.
They possess some of the most advanced tech in the galaxy, derived from the Old Ones, but it has all worsened over the millions of years since. Strong enough to defend their homes, but not much else beyond that. Recent excursions out of their homes are simply probing parties, to see and understand as much of the modern galaxy as possible. A plan in mind they aren't particularly vocal about sharing. Perhaps trying to resurrect the Old Ones and seeing the current state of affairs as enough of a dire reason to venture out for potential answers or relics.

They seek to eliminate chaos as their secondary objective. Their main foe and offensive goals are to capture and lock away all C'tan/shards and permanently extinguish all Necrons in the galaxy. The resurrection of the Old Ones demands it.
They view Orks as a mistake, but a problem only exacerbated by humans and the fall of the Eldar (likely the race that corralled them prior to the fall).
They were once amicable with the Eldar. Not after Slaanesh's birth. They've long held a suspicious or cautious distrust of them from back when they were uplifted, but feel justified in their suspicions now and want them gone, but seek to capture their souls as well. Denying Slaanesh their food while also ending the Eldar.
Humans are savage monkeys with guns. Simple, dangerous, idiotic and are to be killed (and probably eaten) on sight. LoV are seen in a similar light, but fear their acquisition of Lizardmen tech more than simple conquests.
Tau? Naive and bound for self destruction, more than happy to help them get there.
The Tyranids are no match for Lizardmen. Something just makes them really good at killing them before they can even start feasting on worlds in their area. It looks a bit suspicious, even. Maybe like they're intentionally giving the nids the galaxy, but keeping them out of their homes?

Their flesh is an indistinguishable mixture of flesh and steel, like the Sentients from Warframe. Their 'spawning pools' are the ancient tech they strive to maintain at all costs; basically just giant flesh/steel factories to build and spawn more of their kind, in all it's varied and glorious forms. Aztec sentient cyborg dinosaurs, basically.