Stalin knew he would have to worry about the other three, as separate candidates. Trotsky was his main rival. Stalin made Lenin's funeral the centre stage for his plotting. First he tricked Trotsky into turning up to Lenin's funeral on a wrong later date. By not turning up, Trotsky's reputation was damaged. Stalin also used Lenin's funeral to make himself look better by organising the actually funeral, making it look like he should be the one to carry on Lenin's work.
Krupskaya gave Lenin's secret testament to the Central Committee, but Zinoviev and Kamenev didn't it let become public knowledge as not only did it show Stalin as a bad leader, but also Zinoviev and Kamenev didn't look too good in the testament. In addition, they didn't see Stalin as much of a threat and thought they could use him to help defeat Trotsky, who was seen as very good in Lenin's testament.
Zinoviev, Kamenev and Stalin were at the height of the race at the moment. At this moment Trotsky criticised the party calling it too bureaucratic and less democratic. In 1924, Zinoviev and Kamenev built up a ferocious campaign against Trotsky, which questioned his loyalty to Lenin and criticised him about the 1917 revolution, in which he was a bit unwilling to support Lenin's actions. During this time, Stalin thought wisely and stayed in the background.
It wasn't until 1925, when Stalin introduced his "Socialism in One Country" policy, where he wanted to concentrate on Russia