Conversion 9 spd to 10 spd
The first thing to check is that your rear wheel can take a 10 speed cassette. If this is the case, then you can move to step two, which is picking compatible rear derailleurs and shifters. 

A 10 speed Shimano road shifter needs a 7-9 speed mountain/road rear derailleur or a 10 speed road derailleur. A 10 speed Shimano mountain shifter needs a 10 speed mountain shifter. You're proposing a road shifter with a mountain derailleur (both 10 speed) which won't work. 

Once you know the cassette size you want to deal with, you can choose the appropriate model of rear derailleur (long cage, medium cage, short cage, for example) by looking at the required capacity.