There are some circumstances when you don't need probate but the rule is "if in doubt, do it!". (Some organisations, such as banks, will accept a death certificate under certain circumstances but others might not. Obtaining a 'grant of representation', through probate can make things far, far simpler).
Obtaining probate is usually a doddle. It's all explained here:
http://hmctsformfinder.justice.gov.uk/courtfinder/forms/pa002-eng.pdf
Here's the form you'll need to fill in:
http://hmctsformfinder.justice.gov.uk/courtfinder/forms/pa001-eng.pdf
which is made easier by reading the guidance notes here:
http://hmctsformfinder.justice.gov.uk/courtfinder/forms/pa001a-eng.pdf
It will cost you £215 (plus a few multiples of 50p for additional copies of the grant - make sure that you get one for every organisation that might ask for one; it's best to order a few spare copies):
http://hmctsformfinder.justice.gov.uk/courtfinder/forms/pa003-eng.pdf
You'll need to swear an oath (or if, like me, you're an atheist, 'affirm') at one of the venues listed here:
http://hmctsformfinder.justice.gov.uk/courtfinder/forms/pa004-eng.pdf
If you've got any problems, my own experience is that the staff at probate registries are
exceptionally helpful. There's also a helpline available on 0300 123 1072 (M-F, 0900-1700).