How can I see Kurtosis in a box plot? How do I know whether a distribution is leptokurtic or platykurtic by only having the box plot?

Higher kurtosis is indeed indicated by outliers in a box plot. However, it is not the proportion of outliers that determines kurtosis. Instead, the leverage exerted by the outliers (as determined by larger $|z|$-scores) precisely determines kurtosis.  So you can have fewer outliers, but with more extension, that also results in higher kurtosis. Precise statements of this fact are given here and here.
But the box plot does not correspond as precisely to kurtosis as does the normal quantile-quantile plot: There is a direct mathematical and visual connection between the normal q-q plot (and its detrended version) and excess kurtosis that is explained here.
Those who are still promote the incorrect "peakedness" interpretation of kurtosis might suggest that boxplots are inappropriate because they do not show the peak clearly. Here is a counterargument: In a boxplot of 1000 random beta(.5,1) values, shown below, there are no outliers, correctly suggesting low kurtosis (this distribution is less kurtotic than gaussian). But this distribution is infinitely peaked, which cannot be discerned from the boxplot.
Peakedness does not determine kurtosis, and vice versa.

