You'll run towards the bar on autopilot (you can set him off yourself by moving the analogue stick or pressing a run button). Tap the jump button as soon as his head passes the 'i' of 'Konami' which is part of the background graphics (if you press it too late he'll either run straight in to the mat or you'll knock the bar off as you go up). Let him rise and when he leans back you can tap the jump button a few more times to nudge him closer to the bar. As soon as he flips his legs up, hold the jump button to move him all the way over to the right.
Not many people realise that if you run as soon as you've pressed jump for the first time then he'll go a lot higher which makes the jump so much easier and is how you get the world records (2.51m is my best so far!)
To get the achievement you'll need to clear 7 jumps (2.15, 2.25, 2.30, 2.35, 2.36, 2.37 & 2.38) and you can afford to fail twice as you'll get 3 attempts to get there.
Fast forward to 4.33 on the video below to see somebody doing the High Jump although he stops before reaching 2.38 but you'll get the idea.