Yes, you should inform your colleague that you are looking for another job. You must have heard the saying that “people get the best job through networking”. Informing your colleague of your job search can make them help you with useful information that can lead you towards getting the new job.
However, you need to be careful on whom you disclose such information to, some colleagues may go behind your back and inform your boss, if your boss finds out, he or she may take it personal and see this as lack of loyalty to the organization and may take steps towards terminating you.
Therefore, be very careful, take advantage of your chances and make sure you can trust this colleague with such information.
This depends entirely upon your contract. Read this carefully, including the small print. If there is nothing about such disclosures, then I would expect your searches for other work to be your own private business. On the other hand, if you have told peers in your workplace, you risk this information getting back to your boss.
It's always better if he hears such matters from you rather than second hand or from overhearing gossip. If no-one knows your plans in the workplace, then carry out your searches and only when successful should you tell anyone there at all, and make sure the boss is the first to know not the last.
Do things properly and ensure you are giving the correct term of notice. If relevant, thank the boss for the opportunity to work for the firm and state what has been valuable to you.