So I have the US's (America) index chart of S&P 500. Can I look at that US's chart and trade S&P 500 on our NSE?
First of all, I do not trade this, so it is all guesswork.
As long as you are taking medium term positional trades, it should work out. However, if you are planning day trading based on US charts, it may not work that well because the markets keep on moving even when the exchanges are closed and those movements (caused by events such as greece, or libya or XYZ) will not show up on those charts when you start trading. However, for a positional call targeting profits in the next few days / weeks, these intraday variations should not matter that much.