We're currently sitting on the 8th Generation of Intel's Core I series of processors. They also have a number of SKU's per series among the Core i3, i5 and i7 range. What is the SKU for your intended processor? What processor do you currently have?
For the sake of relevance, make sure your motherboard BIOS is up to date.
Asus P8H61-M LX3 R2.0 will support the following CPU. Some CPU will require bios update first with your old CPU before it will work with new CPU. LGA1155 socket is several years old so don't buy new, buy used one on ebay.
Here is the CPU Support List for your motherboard. This lists all the Intel processors that are compatible with your motherboard (to include the 2nd and 3rd gen Core I5 processors). Note that a BIOS update may be required prior to installing some 3rd gen Core I5 processors.
You will have a very hard time finding new cpus for that mobo, it's ancient in regards to new generations. The 'cheapest' best option would be a Xeon 1240v2 to Xeon 1270v2. They range about $100 on ebay. The i5-3570k/3770k have roughly the same performance yet cost more, the Xeon being an i7 equivalent.