Astral Claws are my primary army at this point. Love how they work things. I haven't run Tyrant's Legion yet since I've focused more on the Space Marines/CSM side of their army, using the Claws and Red Corsairs. My army is a pretty good split between the silver and the red.
Play to the strengths of the Chapter every time. You have Stubborn, so take some risks in Assault. Huron is a beast against a lot of models, and adds into a CCW Terminator squad amazingly. The issue is that he has to be still for the Ordnance. It can be worth it to start the game outside of a Land Raider Redeemer, fire that pair of blasts and then start moving forward on Turn 2.
Sumatris does good in an Honour Guard squad or CCW Command Squad. Throw in a Banner that provides a spare Attack and you have a good force. I actually use them in a Land Raider Hyperios. Provides some anti air support with the rest.
Obviously bikes are a must. They're ALMOST as good as White Scars. The lack of Hit and Run means you have to be wary of Assaults, but Move Through Cover and such means that you can load up on Grav Bikes. And my favorite thing to do is have Valthex on the field, not as Warlord because his trait is balls, and upgrade a max unit of Bikes with Melta. Throw in the closest thing we can build to Smashfether and the squad is nearly a Deathstar with 2+ Poisoned Bolters, Melta and Powerfists.
And Land Speeders become your new best friend. Two Multimeltas on a pair of speeders makes for a great Alpha Strike. If your opponent prevents that, swap to Outflank for a late in the game tank killing unit. Alternatively, and I've only tried it once so don't know if it will keep working, but Typhoons and Multimeltas make for a significant threat on the table. Expensive for what they are, but I enjoyed using them and will do so again.
In Tyrant's Legion itself, it's a bit more difficult. The Bikes are expensive as hell, the only speeders you can get are Chapter Support, and the actual Guardsmen are gimped. Fun, but not competitive. Taking a few Auxila troops and then a Formation from the vanilla codex is a good option since a lot of those Formations are very powerful now, but sticking to just that army list is difficult. Not impossible, especially if you're just looking to have fun with the army, but not having access to most of the Special Characters is a letdown. Huron becomes nearly off limits because he can't take a Terminator squad with him unless you house rule the current army list to include the old rule 'Tyrant's Shield' from the physical pages of IA:9 that allows a single Elite choice alongside Huron. Reduces his effectiveness when he's slowing down an entire friggin squad and can't make Sweeping Advances or be in a Rhino.
Mostly, just treat it like a Guard Regiment with a handful of Marines. Chimeras, Hellhounds, Leman Russ and Concripts are the bread and butter. Artillery is a focus, if you build a list around it and work at that, but then you're missing out on the things that make the Astral Claws great...