|Don't buy from eBay, especially if you're not familiar with computer specs.|
There are numerous types & models of CPUs (processors) available from both AMD & Intel - single core, dual-core, triple-core, quad-core & hex-core. The advertised CPU speed is always based on a single core. They are NOT to be combined. For example, if you see a 3.0GHz quad-core CPU, do NOT multiply it by 4 & think it's equal to 12.0GHz. It does NOT work like that. Whoever is claiming to have a system that runs at 9.6GHz is either clueless about CPUs or is deliberately trying to mislead the buyers.
As for HDMI, it's a combination digital graphics/audio interface. Basically, it's just video & sound transmitted thru a single cable. So you will need a system that has a video card that fully supports HDMI & your TV must fully support it as well. Some only transmit/receive video thru the HDMI cable, therefore a separate audio connection would have to be made.
For gaming, you need a decent CPU, lots of RAM, & decent graphics. Generally, integrated graphics isn't going to get the job done. That means the system will need an actual video card. There's a huge range of cards in terms of both price & performance, & the 2 usually go hand-in-hand.
The specs that you listed are incomplete. No motherboard, graphics card or power supply is listed & the RAM & HDD descriptions you provided are incomplete as well.