Manually install drupal in your hosting account is not a complex process, basically it involves three main steps, below describe the installation process in detail:-

Create MySQL Database
Before doing anything, prepare your hosting account with the necessary requirements for running the drupal installation script. First, you should create a MySQL database that will be used by drupal to store it's database schema and data of your drupal website when it goes live later. If you are unsure of the MySQL database creation process, please check create MySQL database and user. Once you have created the needed MySQL database, the database user and grated access for the user to the database, you are ready to proceed to next step.

Download and extract archive
After preparing the MySQL database, you can now download drupal from the official drupal website, be sure to read through the page info and select the latest stable version to download.

Doing with SSH Access
If you have SSH access then you can easily get the file downloaded and extracted in your hosting account, you should first SSH to your hosting account shell prompt, and issue the following command :-
cd public_html
wget path_to_the_drupal_file
tar -xvzf the_download_drupal_file

Doing with FTP
If you are much more familiar with transferring files through FTP, then here are the steps :-

1. visit and download the latest stable version of drupal.
2. note that the compressed file is in standard Unix tar.gz format, be sure you have a compression tool to uncompress files of this format.
3. Uncompress the files in your computer.
4. FTP upload the extracted folder and files into the 'public_html' folder in your hosting account.

Run Drupal installation script
After all files have been extracted and placed in your hosting account 'public_html' folder, you are ready to execute the drupal installation script, this script will set the base URL, connect Drupal to the database and create necessary tables in the database.
To run the installation script, open up a new browser window, enter the following URL :-

http://www.yourdomain.com/path_to_drupal_files/

Note: the installation process require write access to the sites/default/settings.php file, be sure to chmod 666 before running the installation script AND set it back to 644 after the installation process. If the file is not yet present (likely if you are doing a new install), set the folder sites/default permission to chmod 777, this will ensure the installation process will be able to create the settings.php file for you. Make sure you remember to set the folder permission back to chmod 755 after the intallation.

Note 2: drupal requires slightly more memory to function (especially when you are having lots of modules) and you may encounter php memory limit error like the following :-
Fatal error: Allowed memory size of 8388608 bytes exhausted (tried to allocate 46080 bytes)

General PHP default setting for memory_limit is 8M, and your drupal installation may require higher (more needed if you install more modules), you can increase the limit by adding the following line in your .htaccess file that is located in /home/username/public_html/path_to_drupal_files/.htacess file, there should already be a .htaccess file in the folder given by drupal, just add the following line in the appropriate section :-

php_value memory_limit 16M
You will be presented with the Database Configuration screen in the beginning of the installation process, be sure to enter the correct info :-

Database Name : username_dbname
Hostname : localhost
User Name : username_dbuser
Password : dbuser_password