There are a few different ways to calculate this but the most obvious is to simply take an asset or portfolios return and subtract the return on its benchmark or index. A larger figure remaining shows the asset or portfolio had much greater return than the index.
I would note that alpha is not always a desirable thing. Consider an investor who, in order to reduce the risk of their overall portfolio of assets, places a portion of their money into an index fund which has the stated objective of tracking the utilities sector. Large returns in excess of the utility sector may put additional money into the investors pocket but they may also come at the cost of increasing the investors total risk – the exact opposite of what the investor wanted.