Starting off simply:

"I like cats." He said.

That is incorrect because the sentence and him saying it are one complete thought, and the period separates them into different thoughts. It should be:

"I like cats," he said.

…because it combines the two, which go together. The comma merely lets you know that he isn’t talking anymore and that it’s moving onto a different part of the single thought. The comma is necessary because this:

"I like cats" he said.

…is just as incorrect as the first example. 

Adding different punctuation:

"I like cats!" He said.

Although there’s an exclamation mark there instead of a period or comma, capitalising ‘he’ is incorrect because it tells the reader that you’re moving onto something else when it’s still one thought.

"I like cats!" he said.

…is the correct example, and the same thing goes for question marks.

Actions and dialogue:

He raised a hand, “I like cats.”

The action and the dialogue have nothing to do with each other; because they’re different thoughts, they should not have a comma between them. The correct example is this:

He raised a hand. “I like cats.”

In the event of using action to separate two parts of dialogue, only use commas if a sentence is broken up, not if the action goes between two separate sentences. Both are correct:

"I," he paused, but then continued, "like cats."

"I like cats." He shrugged. "Is that a problem?"

Description and dialogue:

"I like cats," his voice held a certain sereneness to it that his persona usually lacked.

The words are not his voice, and thusly the two separate thoughts should be parted by a period, like so:

"I like cats." His voice held a certain sereneness to it that his persona usually lacked.

Dialogue tags (he said, she asked, they exclaimed) are the only things that get commas, and those tags can be descriptive as long as they’re still tags. The following are all correct:

"I like cats," he murmured, voice holding a certain sereneness to it that his persona usually lacked.

He murmured to her, slowly, softly, “I like cats.”