How do I approach them to say 'I have a job but would love to work for you' without making myself look flaky and unreliable?
You can't. Accepting a job while continuing to interview for other positions is a text-book case of an unreliable employee. When you accept a non-seasonal job offer as an employee (as opposed to a contractor), the typical expectation is that you'll stay with that company for at least 1 or 2 years.
You'd be acting in bad faith, disrespecting the company that hired you as well as the other candidates for the job that you pushed out of the running. Any hiring manager that finds out that you've done this will wonder whether you'll do the same after he hired you and most will choose not to take the risk.
If you still want to go through with this, don't mention it to any other employers and don't mention to your current employer why you're leaving or for which company. Frankly, I'd consider leaving your new position off the web for a while as well to avoid (irrationally) spiteful managers.
But my advice is to tough it out in your current position for at least a year, preferably longer to avoid having an instance of job hopping on your resume. The damage this can do to your reputation should not be underestimated.
For a similar perspective on this, have a look at this post by Alison Green on Ask a Manager (emphasis mine):
Now, some people will tell you that it’s fine to renege on that commitment because your employer could fire you or lay you off at any time, or even rescind your job offer before you start. But the reality is that rescinded offers are rare. And very few employers continue interviewing candidates after they make a hire just in case someone better is out there. It’s highly, highly unlikely that your new employer is continuing to look at candidates for your role and that they’d boot you if they found someone better. That just isn’t how it normally works.
And if you think about how pissed off you’d be if that did happen, it might help you look at this differently. You made a commitment, they made a commitment, and you should act in good faith. Because you want to be someone whose word has meaning and who operates with integrity.
Source: should I keep interviewing after I already accepted a job offer?
Caveat: all this assumes that your current job is a regular long-term position and not a short-term contract that both parties can end at any time. If that's the case, you just mention that it's a short-term job with a notice period of X days/weeks.
