You can log out of your account by going to Settings > Your Account > Log Out.
After you log out of your account, your messages in your chats will be cleared.
You'll need to log back in the next time you want to use Kik. You won't get notifications for your Kik messages until you've logged into your account again.
When you log in again, your contacts and list of chats in your main chat list will be there as you left them before you logged out. You'll see Nothing yet as the message preview for a chat in your main chat list if there are no new messages in that chat.