Hop on the Picadilly Line at Heathrow and take it to the Green Park tube station (this will take about 40 minutes). If you have time to head over to Jermyn Street to buy a few shirts, by all means do so. If not, transfer to the Victoria Line at the Green Park station and go North. The third stop is Euston Station (less than 10 minutes from Green Park) where you can catch the train to Liverpool. The tube portion of the journey shouldn’t be more than about 5 GBP. Enjoy!

(Another option is to take the Picadilly Line to the Leicester Square tube station and change to the Northern Line, go North on the Northern Line and that will also get you to Euston Station. Six of one, half dozen of the other.)