While an Intel CPU with the HD 4000 will be more powerful than your current CPU (both the CPU itself and the integrated graphics), the Intel HD 4000 has rather weak performance. The Intel HD 4000 has about 85% - 90% of a Radeon HD 5550 graphics card. Pretty impressive for an integrated graphics solution especially coming from Intel, but it is not the ideal solution.
The Core i5-3570k costs around $220. You will get much better performance buying a $220 graphics card since you current CPU is pretty good. The Radeon HD 6670 is a decent card if you are trying to conserve money. However, I recommend stepping up to the Radeon HD 7750 which you should be able to find for $90 with a rebate or $100 w/o a rebate. It is a little more expensive than the Radeon HD 6670, but you get better performance per $1 than the HD 6670.