Try uname -m. Which is short of uname --machine and it outputs:

x86_64 ==> 64-bit kernel
i686   ==> 32-bit kernel

Otherwise, not for the Linux kernel, but for the CPU, you type:

cat /proc/cpuinfo

or:

grep flags /proc/cpuinfo

Under “flags” parameter, you will see various values: see “What do the flags in /proc/cpuinfo mean?”
Among them, one is named lm: Long Mode (x86-64: amd64, also known as Intel 64, i.e. 64-bit capable)

lm ==> 64-bit processor

Or using lshw (as mentioned below by Rolf of Saxony), without sudo (just for grepping the cpu width):

lshw -class cpu|grep "^       width"|uniq|awk '{print $2}'

Note: you can have a 64-bit CPU with a 32-bit kernel installed.
(as ysdx mentions in his/her own answer, “Nowadays, a system can be multiarch so it does not make sense anyway. You might want to find the default target of the compiler”)