That sempron is a low-end CPU, monocore. Plus 1gb ram and you have integrated graphics there. You will get several troubles to play 2009-2012 games whit this machine. You should get a new sistem, like a dual-core, 2gb ram (min), whit a 1gb DDR3-DDR5 video card.
Your CPU will move forward into a new AM3+ motherboard - and the memory will not (requires DDR3). There is a good likelihood that the PhII 560 X2 CPU will unlock on a new motherboard (better than 50%) going forward.
Your new video card will be limited by your current power supply. A Radeon HD 7750 should work for you if power is an issue starting around $120. At $100 or less, the Radeon HD 6770 1GB.
You can spend more on your video card is yah got the juice for it
If you check on the side of the box these games come in, or go to web sites, you can find the minimum system requirements to run the game. Minimum means just that, the game will just run, not fast, you cannot turn up detail level, the game will just run enough to play it with everything on "low". Your system is a very weak system for today's hardware demanding games. If the game starts but shuts down immediately, this is usually a sign that your video hardware does not support a high enough version of Direct X, a high enough version of Pixel Shader, or maybe even not enough video RAM. And it is also a good indication that even if you could get the game to run, its going to be so slow it will be unplayable.