I'd second the suggestion of buying tickets directly from Avanti, but I think it is actually cheaper to upgrade to Standard Premium on the train than buy in advance. Any Standard ticket can be upgraded, no matter how cheap, and I believe an upgrade costs £20 between London and Liverpool.

(I've never actually upgraded myself, as for my regular trips between Cardiff and Edinburgh it is almost the same price to buy First Class split tickets, but I've heard on-board announcements that the most you pay to upgrade is £25.)

Edit: I now have the actual facts at my fingertips.

Standard Premium | Train Travel | Onboard |Avanti West Coast

The cost of an upgrade between London and Liverpool is £30. You can buy in advance or pay on the train (subject to there being space available). Note, however, that if buying in advance the only option available is an Advance ticket, which limits you to a specific train. If you want flexibility, the better option is to buy an off-peak ticket and pay £30 on the train.