How can I update gcc 5.3 to 6.1?

You can install GCC 6 by adding the ubuntu-toolchain-r/test PPA. To do so, run the following commands:

sudo add-apt-repository ppa:ubuntu-toolchain-r/test
sudo apt update
sudo apt install gcc-6

You can verify that gcc-6 is installed by running gcc-6 --version and the output should say gcc-6 (Ubuntu 6.1.1-2ubuntu12~16.04) 6.1.1 20160510.

As suggested by Mohamed Slama, if you want to further change the default GCC and G++ to the latest versions, install g++-6 with

sudo apt install g++-6

and then run

sudo update-alternatives --install /usr/bin/gcc gcc /usr/bin/gcc-6 60 --slave /usr/bin/g++ g++ /usr/bin/g++-6    

If you want to build it from source (which I recommend as you can for example make a cross-compiler, etc.) download the source from a mirror.

Then extract it with:

tar -xvf gcc-6.1.0.tar.gz

After that change directory to there:

cd gcc-6.1.0

Then create build directory and cd to it:

mkdir build
cd build

Then configure the makefile (--disable-multilib means to not build libraries for cross-compilation):

../configure --enable-languages=c,c++ --disable-multilib

If you ran into errors due to missing required libraries or other prerequisites: (Credits to this)

./contrib/download_prerequisites

And then build it:

make -j 8

This process may take some time and after done invoke this:

sudo make install

That's it!