Update:
Often people want the most recent version of gcc, and devtoolset is being kept up-to-date, so maybe you want devtoolset-N where N={4,5,6,7...}, check yum for the latest available on your system). Updated the cmds below for N=7.

There is a package for gcc-7.2.1 for devtoolset-7 as an example. First you need to enable the Software Collections, then it's available in devtoolset-7:

sudo yum install centos-release-scl
sudo yum install devtoolset-7-gcc*
scl enable devtoolset-7 bash
which gcc
gcc --version

I don't think it's possible to install GCC-5 on CentOS 5, its glibc version (2.5) is way too old.

For CentOS 6, use the following instructions:

• http://en.librehat.com/blog/build-gcc-5-dot-2-on-rhel-6/
• http://www.linuxfromscratch.org/blfs/view/cvs/general/gcc.html

Or you can use Docker with the following Dockerfile:

FROM centos:centos6

RUN yum install -y gcc gcc-c++ wget xz libgcc glibc-devel glibc-headers

# Run some tests
RUN gcc --version && \
    g++ --version && \
    which gcc && \
    which g++
RUN mkdir ~/tests && \
    cd ~/tests && \
    echo '#include <iostream>' > main.cpp && \
    echo 'using namespace std;' >> main.cpp && \
    echo 'int main() {' >> main.cpp && \
    echo '    cout << "Hello world!" << endl;' >> main.cpp && \
    echo '    return 0;' >> main.cpp && \
    echo '}' >> main.cpp && \
    g++ main.cpp -o main && \
    ./main

# Download and compile GCC-5
# http://en.librehat.com/blog/build-gcc-5-dot-2-on-rhel-6/

# Download and extract source code
ENV gcc_version "5.3.0"
RUN wget --no-verbose \
        http://ftpmirror.gnu.org/gcc/gcc-${gcc_version}/gcc-${gcc_version}.tar.bz2 && \
    tar xf gcc-${gcc_version}.tar.bz2
RUN wget --no-verbose \
        https://gmplib.org/download/gmp/gmp-6.1.0.tar.xz && \
    tar xf gmp-6.1.0.tar.xz && \
    mv gmp-6.1.0 gcc-${gcc_version}/gmp
RUN wget --no-verbose \
        ftp://ftp.gnu.org/gnu/mpc/mpc-1.0.3.tar.gz && \
    tar xf mpc-1.0.3.tar.gz && \
    mv mpc-1.0.3 gcc-${gcc_version}/mpc
RUN wget --no-verbose \
        http://www.mpfr.org/mpfr-current/mpfr-3.1.4.tar.xz && \
    tar xf mpfr-3.1.4.tar.xz && \
    mv mpfr-3.1.4 gcc-${gcc_version}/mpfr

# Compile and install GCC
# "we highly recommend that GCC be built into a separate directory from the sources which does not reside within the source tree"
RUN mkdir gcc-${gcc_version}_build && \
    cd gcc-${gcc_version}_build && \
    ../gcc-${gcc_version}/configure \
        --prefix=/usr \
        --disable-multilib \
        --enable-languages=c,c++ \
        --enable-libstdcxx-threads \
        --enable-libstdcxx-time \
        --enable-shared \
        --enable-__cxa_atexit \
        --disable-libunwind-exceptions \
        --disable-libada \
        --host x86_64-redhat-linux-gnu \
        --build x86_64-redhat-linux-gnu \
        --with-default-libstdcxx-abi=gcc4-compatible
RUN cd gcc-${gcc_version}_build && make -j4
RUN cd gcc-${gcc_version}_build && make install

# Validate the installed compiler
RUN hash -r && \
    gcc --version && \
    g++ --version && \
    which gcc && \
    which g++

# Register new libraries with `ldconfig`
RUN echo "/usr/local/lib64" > usrLocalLib64.conf && \
    mv usrLocalLib64.conf /etc/ld.so.conf.d/ && \
    ldconfig


# Clean out all the garbage
RUN rm -rf ~/${gcc_release} ~/{gcc_release}_build ~/tests

you can create repo from fedora and install gcc 5.1.1, it will be better than compile it.

cat << EOF > /etc/yum.repos.d/Fedora-Core23.repo
[warning:fedora]
name=fedora
mirrorlist=http://mirrors.fedoraproject.org/mirrorlist?repo=fedora-23&arch=$basearch
enabled=1
gpgcheck=0
EOF

then install gcc

yum install gcc --enablerepo=warning:fedora

output:

========================================================================================================================================================================================================
Package                                           Arch                                      Version                                            Repository                                         Size
========================================================================================================================================================================================================
Installing:
gcc                                               x86_64                                    5.1.1-4.fc23                                       warning:fedora                                     19 M
Installing for dependencies:
cpp                                               x86_64                                    5.1.1-4.fc23                                       warning:fedora                                    8.5 M
glibc-devel                                       x86_64                                    2.22-2.fc23                                        warning:fedora                                    909 k
glibc-headers                                     x86_64                                    2.22-2.fc23                                        warning:fedora                                    497 k
isl                                               x86_64                                    0.14-4.fc23                                        warning:fedora                                    490 k
kernel-headers                                    x86_64                                    4.2.0-1.fc23                                       warning:fedora                                    992 k
libmpc                                            x86_64                                    1.0.2-4.fc23                                       warning:fedora                                     55 k
mpfr                                              x86_64                                    3.1.3-1.fc23                                       warning:fedora                                    213 k

Updating for dependencies:
binutils                                          x86_64                                    2.25-13.fc23                                       warning:fedora                                    5.6 M
glibc                                             x86_64                                    2.22-2.fc23                                        warning:fedora                                    3.6 M
glibc-common                                      x86_64                                    2.22-2.fc23                                        warning:fedora                                     11 M
libgcc                                            x86_64                                    5.1.1-4.fc23                                       warning:fedora                                     82 k
libgomp                                           x86_64                                    5.1.1-4.fc23                                       warning:fedora                                    146 k

Transaction Summary
========================================================================================================================================================================================================
Install  1 Package  (+7 Dependent packages)
Upgrade             ( 5 Dependent packages)

Total download size: 52 M
Is this ok [y/d/N]: y

Why do you preferred on gcc 5.2?

# gcc --version
gcc (GCC) 5.1.1 20150618 (Red Hat 5.1.1-4)
Copyright (C) 2015 Free Software Foundation, Inc.
This is free software; see the source for copying conditions.  There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR  PURPOSE.

I've confirmed that you can upgrade gcc from the default version 4.8 on centOS 7.

First, we need to install "Software Collections" in order to access some of the community packages including gcc v7

• sudo yum install -y centos-release-scl

Next, we want to install a developer toolset. Depending on your needs, you may want a different devtoolset. Here I'm targeting 7:

• sudo yum install -y devtoolset-7

Finally, you'll want to change over to gcc 7 as your default, launch a new shell session with the scl tool:

• scl enable devtoolset-7 bash

CentOS 8 already comes with GCC 8.

On CentOS 7, you can install GCC 8 from Developer Toolset. First you need to enable the Software Collections repository:

yum install centos-release-scl

Then you can install GCC 8 and its C++ compiler:

yum install devtoolset-8-gcc devtoolset-8-gcc-c++

To switch to a shell which defaults gcc and g++ to this GCC version, use:

scl enable devtoolset-8 -- bash

You need to wrap all commands under the scl call, so that the process environment changes performed by this command affect all subshells. For example, you could use the scl command to invoke a shell script that performs the required actions.

Check out the 'Supported Libraries' section in https://gcc.gnu.org/wiki/InstallingGCC. I was successful in building gcc 5.3 on Centos 6.5 by running

./contrib/download_prerequisites 

mentioned in the above doc.