I have UK travel visa (Standard Visitor), visited it once. Do I have to declare the intention (e.g. tourism) for additional visits to border control? If yes, how should I do that properly?
You don't have to do anything. Your visa/entry clearance remains valid until you violate its terms or it has expired or you no longer satisfy the criteria e.g you have subsequently committed crimes which make you ineligible, etc.
For example, Paragraph 320 of the UK Immigration Rules lists a number of grounds for refusal of entry clearance or leave to enter. You can verify yourself if you fall afoul of those rules.
Read the Visit guidance
At your landing interview they will determine whether to let you in or not as they do with all aliens who presents themselves at the border.
Do I have to declare the intention (e.g. tourism) for additional visits to border control? If yes, how should I do that properly?
You may be asked to declare your intention. To do it properly, you should respond to the immigration officer, who will ask "what is the purpose of your visit" or something to that effect.
So having tickets/hotel and similar proofs is enough?
It should be. In general, you should have the same sort of evidence supporting the premise of your trip as you would present for a visa application. The usual advice is that you should present this evidence if you are asked for it rather than volunteering it up front.
The general principle is that you should avoid volunteering information, other than presenting your passport when you step up to the desk, but you should not hide anything if asked. Above all, never lie. Immigration officers do not like being lied to.