How to successfully approach agents
Remember what an agent wants you for, essentially, they want to make money out of you. You are two businesses going into partnership to create money. So when you approach an agent, you need to come across in a way that shows you understand this. That means keep it professional, research how they like to be contacted and no gimmicks.
Any application to an agent should include a headshot, C.V and cover letter as standard. It's best to also include showreel links/footage too because they will want to see their 'product' in action to evaluate how good it is and whether they think anyone will want to buy it. If you don't have a showreel try and write a month or so before a show you are doing so that you can invite them to see it. An agent will need to see your work before they sign you.
Make sure your headshot is professional, don't cut corners here. It's your calling card, your shop window, it is extremely important. A bad headshot tells the agent that you don't do things properly and don't care. Why would they risk their reputation on you? Spend the money and get it done right.
There are whole topics on C.V's so I won't go into it again here, but don't include amateur stuff and keep it to one page. Mention anything that sets you apart from others (boxing skills, accents, singing etc) but don't lie. If they take you on because you say you sing to a high standard when you don't, they will be very angry when the MT casting director you have just auditioned for calls and tells them you aren't very good.
A good covering letter can really make the difference between getting a meeting and not. Keep it short, to the point and professional. Don't speak badly of previous agents, don't come across desperate or dreamy. Highlight anything note worthy like high profile jobs or skills that are especially impressive and if possible, show you have done your research by mentioning work you have seen their current clients doing ("I thought John Doe was fantastic in The Tempest last month"). Having said all that, try to inject a little personality if possible, don't be dull.
If you are asked to a meeting it doesn't guarantee you will be signed, so turn up well rested, looking good and in a positive frame of mind. Have answers to possible questions prepared and have questions ready to ask back if the opportunity arises. It can be uncomfortable for them if you treat a prospective agent as a God so be yourself, don't be afraid of having slightly different opinions on things but obviously don't be argumentative either!