Since you're staying at 10 speed (ie, 10 different gears on the rear cassette) then you don't need a new shifter on the handlebars. That's a wise plan. If you got an 11 speed cassette you need also a new shifter, which gets expensive.
Your rear derailleur should be okay for the width, but the problems come about trying to clear that top gear. Its hard to tell if you'll get away with 36 tooth big gear, or whether your rear mech can only cope with a 34 or 32. My road bike takes nothing larger than a 28 tooth.
Do you have a short/medium/long cage rear derailleur?
As Batman says, you need a new 10 speed chain when you change cassettes. Worn chains slip on new cassettes and drastically increase wear.
Another thought - if you're not pulling top gear a lot then you can forgo the 11 tooth little cog and have a 12 or even a 13 tooth small gear.