NEW YORK, NY - Don't use company equipment. Your corporate laptop, smartphone, and wifi are all off-limits for your confidential job hunt. Even if you clear your browser history religiously, there's no way to be sure that Big Brother isn't watching you. Put in 100% at your current job. Don't leave before you leave. Tempting though it may be to disengage completely, if you're still collecting a check, you should put in the work.
Don't lie, at least, not more than you have to. Everyone has had the fake dentist appointment that covers a job interview, but unless there's no other way to get time off, don't lean on these excuses. It's far better to take a personal day, if possible, and keep the fibbing to a minimum. Be careful on social media. You'll have to network and using LinkedIn is crucial, but remember to be covert, to avoid the risk of your current boss becoming suspicious. Don't change all your online headlines and bios in such a way that your boss will be forced to recognize that you're looking for something new.
As for posting your resume, you need to be careful. Some job boards and resume posting services will allow you to hide portions of your resume to maintain confidentiality. Before you post your resume to a site, check out the features available. Don't post your current company phone number or email address for your contact information.