Ok here we go on how to compile and install a new kernel :
First the dependencies for all of this :
sudo apt-get install build-essential makedumpfile
Plus the dependencies for the specific kernel, for a kernel which is in a repository you can do this :
sudo apt-get build-dep linux-image-"kernel number"
For your kernel, I don't know, maybe they are the same maybe they are differents you should search info on Google for that, it may depend on how you configure your kernel.
After this I suggest you create a place to work in your home folder :
You can unzip your kernel here or get a kernel from a repository with :
apt-get source linux-image-"kernel number"
Now we want to work in the unzipped folder :
cd linux-"kernel number"
After this you will need a tool to edit your kernel configuration, there are two different tools and to use them you will have to install their dependencies :
For xconfig = sudo apt-get install qt3-dev-tools libqt3-mt-dev
For menuconfig = sudo apt-get install libncurses5 libncurses5-dev
I suggest you use xconfig, which is easier to use. To use it you type :
xconfig = make xconfig
menuconfig = make menuconfig
Configure your kernel as you like and save your configuration. If you don't need it you should disable "Kernel debugging (DEBUG_KERNEL)" under "Kernel hacking" since that will make the compilation less long.
To compile, you type :
CONCURRENCY_LEVEL=3 sudo make-kpkg --initrd --append-to-version=-something kernel-image kernel-headers
Replace "CONCURRENCY_LEVEL=3" by the number of core your processor have +1 that will make the compilation much faster. Replace "--append-to-version=-something" by what you want to append to the name of the kernel.
Wait for a very long time, several hours depending on your computer.
When it is finished and if nothing bad happened you will have two beautiful debian packages created in the parent folder, install them :
sudo dpkg -i linux-image-"kernel number".deb
sudo dpkg -i linux-headers-"kernel number".deb
Don't forget to update grub :
sudo dpkg-reconfigure grub-pc
Reboot on your new kernel and, if everything is fine, you're done.