She says "Talos be with you" when you kill Ulfric.
Hmm?

Nothing strange there. Most Legionnaires still worship Talos, though discreetly (he's an Imperial god, after all). Plus, Alvor said that in his youth, everyone worshiped Talos even though it was prohibited. Rikke is a Nord, likely born in Skyrim, and could be around Alvor's age.
Also, you might want to put that first line in spoilers.

She's a Nord in the Legion who is still (secretly) worshipping Talos despite the ban imposed by the WGC. She and Ulfric (and Galmar) are/were old friends, most likely they fought together in the Imperial military during the Great War. Despite being on different sides in the current conflict, they do still have some positive feelings for each other. More so in Galmar and Ulfric's case I think than Rikke's, but that's a matter of opinion.

This may not be as obvious if you play the Legion questline, however if you play the Stormcloak side when you get to the end Ulfric and Galmar will implore Rikke to just walk away and let them deal with Tullius alone. They don't want to fight her and don't want to kill her.

I didn't kill Ulfric, Tullius did. I refused on the grounds that he wasn't the true villain. And the only reason the Empire doesn't worship Talos openly is because the Thalmor have told them not to...or else.

She's speaking for what she thinks is best for her people. She isn't thinking with her heart, but what she believes will result in the continued existence of Nords and humans in general. This is my opinion anyhow.

Edit: Over-zealousness stamped out

Being an imperial like Rikke (and others like Hadvar) is about thinking in terms of unity, compromise, thinking that strength lies in working together. They aren't in the legion because they support the banning of Talos worship.

yep, who support the banning of Talos Worship is in the Thalmor not the empire or stormcloaks q-

TBH the only time I've noticed her doing anything like "spewing forth" dialogue is when she blurts out her snarky little sotto voce jibes at Ulfric during Season Unending. The rest of the time her VA runs the gamut (IMO) from nails-on-a-chalkboard to "some drowsiness may occur, do not play this questline while driving or operating heavy machinery."

Yeah, I was a bit overzealous there. I changed it right as you mentioned it

But indeed. I believe she's simply wants to continue life without extensive bloodshed as she believe will happen under Ulfric's rule. She has put her faith in the empire to stamp out the Thalmore, but I don't think she gives a damn about the empire itself or it's greater goal. Just protecting her people the only way she knows how.

Some people are outright rebels, some don't like the world shaking up and prefer slow, steady progress. Sometimes the two even have the same goal in the end, but then it becomes a conflict of implementation, not goals. Same stuff happens everyday, with friends, with work. What I hate about this game's storyline is how stubborn and murderous each side is to each other. The most bitter enemies are those who don't share the same goals. Only a fanatic would kill over how something is implemented. edit: whoa, sorry for the soapbox moment. I think I was trying to be relevant somehow.

She is probably a bit older, being a veteran from the Great War and all. But yes, it seems that most Imperials and Imperial alligned people still worship in secret.