You cannot run Hybrid Crossfire using the on-board HD 4200 with the Radeon HD 4770.
Why are you buying a "End of Life" Radeon HD 4770 graphics card? AMD no longer makes them...even the whole Radeon HD 4000 line-up has now been moved to legacy products and only get driver updates every ~3 months (like the older HD 2000 / 3000 cards).
I would look into newer generation cards. They are not too expensive, and use less power.
Speaking of power, a 300W might not be enough power for your entire system + a graphics card. You may need to replace it as well...
I can't go into much more detail...without your budget (how much are you willing to spend for this upgrade).
Phenom II X6 1090T @ 4.1GHz FX-8350 @ 4.8GHz // ASUS Crosshair V Formula 990FX // Sapphire Radeon HD Dual-X 7970 @ 1150/1500 // Thermaltake Frio w/ push-pull (using Antec Formula 7 Nano-Diamond + Cooler Master JetFlo fans) // 8GB (2 X 4GB) G.Skill RipJawsX 2133MHz // Corsair TX850 850W // Corsair Force GT 120GB SSD (OS) // Kingston V200+ 120GB SSD // WD Caviar Black 1TB // Windows 7 Ult x64