Creating Li-Ion charger with step up to 700mA from 500mA?
USB supplies 5V at maximum 500mA. Power-wise this is 5 x 0.5 = 2.5W.

Your charger supplies 4.2V at 700mA. Power wise this is 4.2 * 0.7 = 2.94W.

You cannot get more power out than goes in, so you cannot charge at 4.2V @ 700mA. To increase the current to 700mA you would need to lower the voltage below 4V, which is not really an option for Li-Ion charging. The maximum current you could achieve (assuming 100% conversion efficiency) would be 2.5W / 4.2V = 0.595A. In reality due to losses it would be more like 595mA * 0.9 = 535mA.
So you might as well skip the conversion and just use the 500mA directly.

At 500mA, it will just take a bit longer to charge.  

Li-Ion charging can get quite complicated, but to get to started the MCP73831 is a very simple/cheap charge control IC from Microchip, who make quite a few decent charge control ICs of varying complexity. It is selectable between 100mA and 500mA, and has a pin for LED indicator.
The MCP73837 can switch between a DC apadtor and USB, has more control over charge current, and a thermistor input.
Read both datasheets thoroughly, they give plenty of good information and example schematics.