There is really no at home test that can tell you whether your diamond gemstone or any natural gemstone is real or not. You will need to know what a gemstone is and its cut, color, clarity and carat grades/values in order to know (1). whether it is real and (2). what its value and quality is exactly.
You can only know this with the right tools and with the help of a trained and experienced gemologist. You may get this for free by visiting a reputable jewelry store or pawn shop near you and by asking the store/shop to tell you how much they are likely to pay for your diamond gemstone or piece.
You may only want to use those that would do this for you while you wait. The store/shop will need to do an analysis of your diamond gemstone or piece in order to know how much they can pay for it. You can therefore use this to get a free analysis of your diamond.
In addition to this, sometimes even the jewelry stores/shops may not have the advanced equipments/tools to determine if a diamond gemstone is real or not or may not be able to determine its exact grades or attributes even after doing some analysis of their own. Hence the store/shop may tell you to send your diamond or gemstone to an institute like GIA (Gemological Institute of America) for some further analysis, which is a paid service.
In order for them to know exactly what they have, a lot of end-users may send their diamonds or gemstones to GIA after this for more analysis, if they have gotten some idea from the jewelry store for instance, about the value of their diamond/gemstone (especially if it is an expensive piece).
Any other tests apart from this will most likely not provide you with the real information that you may need about your diamond gemstone or piece.