Download a version of sqlite3, e.g. sqlite-3.3.4.tar.gz, from http://www.sqlite.org.
Move it to the home directory.
To verify the package is there, type
ls
To unpack sqlite3 type
tar zxf sqlite-3.3.4.tar.gz
To enter the newly created directory, type
cd sqlite-3.3.4
To configure sqlite, type
./configure --prefix=/usr
Watch many messages, but no errors.
To compile sqlite, type
make
Watch many messages, but no errors.
To install sqlite, type
sudo make install
and provide the password.
Watch many messages, but no errors.
To go back to the home directory, type
cd
Optionally the tarball and the directory it made can be cleaned away. To do that, type
rm -rf sqlite-3*