By optimizing your LinkedIn profile with keywords, you'll have a higher chance of showing up in recruiters' search results.  They'll be able to find you and see that you have the skills and experience they need for the job opportunity. This is your ideal situation: instead of 

having to reach out to recruiters in your job search, they'll come to you!

Details matter! You may learn from the recruiter's LinkedIn profile that you attended the same university or once worked at the same company during different time periods.  Anything you have in common with the recruiter could be a good way to break the ice.

Use the simple 300-word connection request to introduce yourself to the recruiter.  Mention something that you have in common and that you're interested in learning more about their career field and would love the opportunity to connect.Keep it simple—you don't want to overwhelm the recruiter.

Once you're connected, send a follow-up note thanking them for the connection, mention that the job posting caught your attention, and ask them what's most important to them when considering a candidate for this position.  This is a way to express your interest in the position while politely

asking for their feedback. The key is not to be pushy and frame it in a way where you're trying to make their job easier.