I'm not sure if it was the first or not but one of the first PC's was actually a game system. Atari made a system that played Ping Pong on your TV set. The game destroyed the phophor coating on the TV screen and gave rise to the invention of the "screen saver." Another early model was the Commodor C64 home PC that could play simple DOS games that could be written and distributed on 56 K floppy discs 5 1/4 inches on a side.

To correct the poster above - C64 did not use DOS, it was a custom OS with a built-in BASIC interpreter, with 38K out of 64K available for programs, that could be stored on tape drive, or 180K or 360K floppies, not 56K. The first commercially available home computer was, I think, was a Tandy TRS system in the 70's. Cannot talk about video games as the question is about "home computers". Sinclair made a ZX-80 in 1980, then the ZX-81 in 1981, and this had a whole 1K of memory to work with, with a kludgy membrane keyboard. Commodore introduced the VIC-20 in 1982 with 3.5K of memory and a proper keyboard and tape drive and a cartridge system. Then the industry exploded, with more offerings from Commodore, Sinclair, Tandy, BBC, the PC!, Apple, etc, etc, with computers getting faster and bigger every year.

In some ways, the first home computers were not known for capabilities, because they had very little. Rather, they were the start of an evolution into the hi-tech world. For example, the Commodore Vic-20 was a very weak home computer (especially to todays standards). It had only 3.5K built in memory and data was 99% stored on standard audio tapes. Meaning that if, in general terms, each character takes up a byte of memory and 1024 characters is a K, then you can see how limited it really was. But what it lacked in storage power it gained by being a home computer that had a built in language (Basic) and when you turn it on, it went directly to that programming language. Thus, those who used those first home computers home-learned how to program a computer. And because it had limited memory, how to get the most out of it. Some programs made back then were nothing short of amazing based on those memory limitations. Early home computers (like the Vic-20) actually were "smart" machines with small storage size. Those who graduated to the more advanced systems, later in life, became superior programmers, because they learned their trade on the first home computers. It offered the ability to create customize characters, sounds and colors; and even high-res with the use of additional cartridges. The capabilities were small but these first home computers opened the door to a society that accepted computers. Commodore products, such as the Vic-20 and 64, played a significant role in making this happen.