Installing the ruby2.0 package does not install the ruby2.0 but actually installs ruby1.9.3 instead.

You may wish to consider using a ruby package manager like rvm or rbenv
You can install different rubies and switch between them easily.
You might also want to consider trying 2.0+

Sample output from rvm:
21:59:48 Troll@menomore /home/Troll
$ rvm list

rvm rubies

ruby-1.8.7-p374 [ x86_64 ]
ruby-1.9.3-p125 [ x86_64 ]
ruby-1.9.3-p194 [ x86_64 ]
ruby-1.9.3-p448 [ x86_64 ]
ruby-2.0.0-p195 [ x86_64 ]
=* ruby-2.0.0-p247 [ x86_64 ]
ruby-2.0.0-p481 [ x86_64 ]
ruby-2.1.1 [ x86_64 ]
ruby-2.1.2 [ x86_64 ]

# => - current
# =* - current && default
# * - default

21:59:50 Troll@menomore /home/Troll
$ rvm use 2.0.0
Using /home/troll/.rvm/gems/ruby-2.0.0-p481

$ rvm use 2.1.1
Using /home/troll/.rvm/gems/ruby-2.1.1

$ rvm use 1.9.3
ruby-1.9.3-p547 is not installed.

$ rvm use 1.9.3-p448
Using /home/troll/.rvm/gems/ruby-1.9.3-p448

Get rvm at http://rvm.io/

Install with its famous 1 liner:

$ \curl -sSL https://get.rvm.io | bash -s stable

Have you tried following the way of installing ruby 2.1.0 by RVM?

If you really don't want to use RVM, then use

sudo apt-get install checkinstall

wget -c http://ftp.ruby-lang.org/pub/ruby/2.0/ruby-2.0.0-p0.tar.gz
tar -xzf ruby-2.0.0-p0.tar.gz
cd ruby-2.0.0-p0

./configure
make

sudo checkinstall -y \
--pkgversion 2.0.0-p0 \
--provides "ruby-interpreter"

checkinstall will package the source, making it easier to remove in the future
You'll then need to add the Ruby binaries to your path, by editing the env file:

sudo nano /etc/environment

add /usr/local/ruby/bin

PATH="/usr/local/ruby/bin:/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin"

then run

source /etc/environment

to reload the file, and check your ruby version with

ruby -v

Should I remove the local Ruby before doing this?