Using the instructions from docs.asp.net, for Ubuntu we essentially run:

login to ubuntu
fire up terminal (Ctrl-Alt-T)

run the following commands:

sudo apt-get install unzip curl

curl -sSL https://raw.githubusercontent.com/aspnet/Home/dev/dnvminstall.sh | DNX_BRANCH=dev sh && source ~/.dnx/dnvm/dnvm.sh

sudo apt-get install libunwind8 gettext libssl-dev libcurl4-openssl-dev zlib1g libicu-dev uuid-dev

dnvm upgrade -r coreclr

sudo apt-get install make automake libtool curl
curl -sSL https://github.com/libuv/libuv/archive/v1.4.2.tar.gz | sudo tar zxfv - -C /usr/local/src
cd /usr/local/src/libuv-1.4.2
sudo sh autogen.sh
sudo ./configure
sudo make
sudo make install
sudo rm -rf /usr/local/src/libuv-1.4.2 && cd ~/
sudo ldconfig

Happily, this all ran without a hitch.

References: http://docs.asp.net/en/latest/getting-started/installing-on-linux.html#installing-on-debian-ubuntu-and-derivatives