Toilet Tank Issue, Can I Fix Myself?

My toilet has been running and the water bill was over double what it usually is, so when I looked inside my toilet tank I noticed a constant drip right below the fill valve, and when I flushed, water was spraying out of it all around. What needs to be replaced on this, and is it something I can do on my own? My landlord is taking his sweet time getting around to it, and I'm fairly mechanically-inclined, so I'd rather just do it and get it over with myself.