Typing Polytonic Greek with a UK pattern keyboard (Win XP pro set for extended Greek) | note
UPPER CASE i, y, A ! " $ % ^ & * ( ) _a, _y +a, +i, +u, +w [back space] uk
Greek ῒ, ῢ, `Α ! @ # $ % ^ & * ( ) ᾰ, ῠ, ἇ, ἷ, ὗ, ὧ   Greek
lower case `i, `y, `a 1 2 3 4 5 6 7 8 9 0 -a, -i, -y =a, =i, =y, =v   uk
Greek ΐ, ΰ, ~α 1 2 3 4 5 6 7 8 9 0 ᾱ, ῑ, ῡ ἆ, ἶ, ὖ, ὦ   Greek
UPPER CASE [tab] QI, Qi W E R T Y U I O P {a, {v }a, }i, }y, }v [return] uk
Greek   Ϊ, ϊ ς Ε Ρ Τ Υ Θ Ι Ο Π ᾳ, ῳ,     Greek
lower case   q w e r t y u i o p [a, [i, [y, [v ]a, ]e, ]i, ]y, ]v   uk
Greek   α ς ε ρ τ υ θ ι ο π ᾶ, ῖ, ῦ, ῶ ὰ, ὲ, ὶ, ὺ, ὼ   Greek
UPPER CASE [caps lock]   A S D F G H J K L :i, :y @a, @e, @i, @y, @v ~a, ~e, ~i, ~y, ~v uk
Greek     Α Σ Δ Φ Γ Η Ξ Κ Λ ϊ, ϋ ἁ, ἑ, ἱ, ὑ, ὡ ἃ, ἓ, ἳ, ὓ,ὣ Greek
lower case     a s d f g h j k l ;a, ;e, ;i, 'y, ;v 'a, 'e, 'i, 'y, 'v #a, #e, #i, 'y, #v uk
Greek     α σ δ φ γ η ξ κ λ ά, έ, ί, ύ, ώ ἀ, ἐ, ἰ, ὐ, ὠ ἂ, ἒ, ἲ, ὒ, ὢ Greek
UPPER CASE [shift] | Z X C V B N M < > ?a, ?e, ?i, ?y, ?v [shift]   uk
Greek   > Ζ Χ Ψ Ω Β Ν Μ < > ἅ, ἕ, ἵ, ὕ, ὥ     Greek
lower case   \ z x c v b n m , . /a, /e, /i, /y, /v     uk
Greek   < ζ χ ψ ω β ν μ , . ἄ, ἔ, ἴ, ὔ, ὤ     Greek

See Alan Wood's Unicode Page.
Note: I haven't found any way yet to type the Greek letters combining both upper and lower symbols. i.e. (unicode decimal 8070), (unicode decimal 8082) etc.