From here:

Get the limited API, it's useful for finding out how many skill points a pilot has and it lets you see their alternate characters (alts). Check their employment history and their alts' employment history. If your corporation has vulnerable members such as miners or missioners, you need to check their killboard to ensure that they have no kills against members of their own corporation. Some spies will join a corporation just to pop one of your Hulk pilots without a security hit and joining a corporation makes suicide ganking that officer fit marauder no longer suicidal [Also known as Awoxing -Ed]. Make sure their killboard, employment history, and skills match up with what they are telling you about themselves. If you see a Hulk pilot with a 10:1 kill ratio and perfect Nightmare skills, you should ask him why. If all this checks out you go to EVE Search and run a search with their character name in it, this will alert you of any butthurt people they may have robbed or harmed in the past.

In addition, if you have the API access, check their wallet balance/assets for anomalies. In particular, when you interview them, ask them questions and see whether their responses throw up red flags (expecially if they try to disguise themselves using poor english or pretending to be noobs)

You must check the API keys of every character you let into the corp - unless you want someone to say "oh yeah please accept X because he's my mining alt" and you only find out later that the alt can fly every ship under the sun and has flown them away after their main stole them from the hangar.

Be especially careful of characters that have been sold on the character bazaar. Many spies and corp thieves buy existing characters with a clean history, burn them in a heist, then on-sell them to finance the next character.

Also, have you seen this question?