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The development of noninvasive brain imaging such as resting-state functional magnetic resonance imaging (rs-fMRI) and its combination with AI algorithm provides a promising solution for the early diagnosis of Autism spectrum disorder (ASD). However, the performance of the current ASD classification based on rs-fMRI still needs to be improved. This paper introduces a classification framework to aid ASD diagnosis based on rs-fMRI. In the framework, we proposed a novel filter feature selection method based on the difference between step distribution curves (DSDC) to select remarkable functional connectivities (FCs) and utilized a multilayer perceptron (MLP) which was pretrained by a simplified Variational Autoencoder (VAE) for classification. We also designed a pipeline consisting of a normalization procedure and a modified hyperbolic tangent (tanh) activation function to replace the original tanh function, further improving the model accuracy. Our model was evaluated by 10 times 10-fold cross-validation and achieved an average accuracy of 78.12%, outperforming the state-of-the-art methods reported on the same dataset. Given the importance of sensitivity and specificity in disease diagnosis, two constraints were designed in our model which can improve the model's sensitivity and specificity by up to 9.32% and 10.21%, respectively. The added constraints allow our model to handle different application scenarios and can be used broadly.