How do I know that my profile has been liked?

You have to like the profiles that you like, and if they like you back or already liked your profile, you will have a match and you will be able to chat.

If you want to see who liked you before liking them, this is a paid option included in the Golden subscription.

Where can I see the profiles I liked?

You can only see these profiles if they like yours back. Then you'll have a match and you'll find it in your matches.