Dogs are easy. Cats, not so much. They belong to a secret club, it seems. When you see a photo op they almost certainly reorganize their bodies by the time you get back with the camera.
The best advice I can give is to always keep a camera handy and make sure you have a good light before snapping. A cat is going to be sleepy-eyed when it wakes up, and there is usually a stretch and/or yawn that follows. If you are shooting with a camera phone you may get some movement unless you have great light.
The best time to capture your beauty is shortly before he winks out, or is engaged in watching something that interest him enough to be bright-eyed but sleepy enough to not want to get up to investigate.
Look for bright eyes, relaxed poise, and good light. Catch a cat when he is in deep thought.
In this photo, Skipper is getting ready for his afternoon nap but the birds have peaked his interest. He is distracted long enough to give a quick glance towards the camera. Notice that he has a lovely light falling on his face, which is what prompted me to grab my camera. I took this with a telephoto lens.
There was no great light on Khaleesi but her pose were perfect, and there was just enough light to bring out her blue Siamese eyes. Like Skipper, she was also contemplating a nap. She was in such deep thought she paid the camera no attention. I submitted this photo to the 2018 Workman's 365 Day Calendar. She won first place which make her the January 1 calendar cat.
Don't touch that photo! Copyrighted, you know?