How do I paste the Windows clipboard into my PuTTY session, using only the keyboard?

You can use Shift+Ins to paste text.

From PuTTY documentation:

Pasting is done using the right button (or the middle mouse button, if you have a three-button mouse and have set it up; see section 4.11.2). (Pressing Shift-Ins, or selecting ‘Paste’ from the Ctrl+right-click context menu, have the same effect.) When you click the right mouse button, PuTTY will read whatever is in the Windows clipboard and paste it into your session, exactly as if it had been typed at the keyboard.

From the Putty Documentation

4.10.3 Changing the actions of the mouse buttons PuTTY's copy and paste mechanism is modelled on the Unix xterm application. The X Window System uses a three-button mouse, and the convention is that the left button selects, the right button extends an existing selection, and the middle button pastes.

Windows typically only has two mouse buttons, so in PuTTY's default configuration, the right button pastes, and the middle button (if you have one) extends a selection.

If you have a three-button mouse and you are already used to the xterm arrangement, you can select it using the "Action of mouse buttons" control.

You have to change the default behavior by opening Putty > Windows > Selection > Select "xterm (Right extends, middle pastes)"

In my case I want Ctrl-Ins / Shift-Ins only, thus, my config was:

• Window > Selection
  • Auto-copy selected text to system clipboard > (unchecked)
  • Mouse Paste > (No action)
  • {Ctrl,Shift}+Ins > (System clipboard)

You need this to VIM works:

• Terminal > Features
  • Disable application keypad mode (checked)

Extra:

PuTTY operates just like a normal X terminal... Some commands will output to an "alternate" screen when the xterm terminal type is selected, such as "less" (or "vi"), which dissapears when the program exits.

However, you can disable the alternate screen with:

• Terminal > Features
  • Disable switching to alternate terminal screen (checked)