I have an old mountain bike with a 5 speed rear cassette. The cassette, chain and cable are very worn and I plan to change the all.
I am quite happy to change them like for like, but 5 speed cassettes are hard to come by so I plan to change to to a 6 or 7 speed.
My questions are;
1) Will I need a different chain length ?
2) Will I need to change the rear derailier ?
I know I will have to change the shifter and will probably go from twist to lever type.
As Porkypete have said you are likely to have a freewheel rather than a cassette.
New 5 speed freewheels are by no means hard to get. For example see
these and
this. I believe Halfrauds sell bucket loads of them (for £13 a pop, own brand ones to boot!) on a daily basis e.g., many new (cheap) bikes only have 5 speeds.
However as you can see from the sjs selection yours might or might not be
"standard" iso threaded, or have 5.5mm pitch. It is worth checking.
Assuming you are replacing the old with a new 5 speed freewheel (thereby sidestepping all potential hub / frame OLD / shifter issues), the answers to your questions are:
1) As others have said you only need a longer / shorter chain if the largest new cog is larger / smaller than the old one. But since you are getting a new chain, if I were you I would ignore what is there and have the ideal length
calculated.
2) Assuming it works you will need to change your rear mech only if a) the largest new rear cog is larger than the largest old cog AND b) the rear mech's capacity is exceeded in terms of either max cog size allowed or max chain wrap capacity. But since in practice such max limits are neither an exact science nor is manufacturer's spec reliable, if I were you I would try to see what the existing mech can do.
This is the best article I have ever seen on how to squeeze the maximum capacity out of a rear mech.
Hope it helps.