I'll preface this with the following: Since you mention essay I'm assuming this is school-related? If so, do not - under any circumstances - say that Obamacare increased insurance prices. Because that is (virtually) impossible to prove unless you can somehow get a letter from some insurance company executive outright stating that. And without some solid proof, a biased professor will just slap a "correlation does not equal causation" stamp on that section of your paper and start taking points off.

That said, Obamacare did increase costs. And while you can't say that in an academic setting what you can say is something like "Obamacare did nothing to address increasing insurance costs and in a number of cases those costs have increased faster than pre-Obamacare". Because that is easily provable by looking at trends over time. One source, for example, is right here Health care costs in the US have been going up faster than inflation for decades, and if you look at the graphs towards the bottom of that link you'll notice there is no slowing down in the growth and a suspicious spike from 2010-2011, almost as if some bill was signed into law in 2010 which impacted things greatly.

If you want actual firm data, probably the best way to do it is how Ahaus667 suggested - grab single HMO plans from several health insurance companies from several years over the last few decades and compare prices over time. Only potential issue there is that what the plans cover will have changed, in large part due to a number of laws - most notably Obamacare - mandating more things be covered and more people be covered. But, there's not much you can do about that.