Building towards a climax requires also your melody to increase tension which is not only possible by “going higher and higher”. When analyzing melodic structures that build towards a climax, usually several characteristics of the melody push it towards the climax. Apart from the already mentioned “higher pitches” which obviously create the feeling of a buildup, successful melodic buildups start to create tension also on a rhythmic level by using more rhythmic activity. It also helps tremendously is to increase the interval sizes towards the climax.
One of the masterful examples of how to build up melodic tension towards a climax is John Williams’ “Leia’s Theme”. Have a listen to the string melody close to the big climax and how much it increases subjective speed by using smaller rhythmic values and how much the melody becomes more and more restless by spanning larger and larger intervals. Analyzing this melody shows that every single property of the melody “pull on the same end” to bring it to the climax.
And this is just the melody. If you analyze the whole passage, everything in the writing (dynamics, harmonic progression, orchestratration) serves the one purpose to bring it to the climax. All elements in the music writing masterfully follow this one purpose.