Learning high-quality representation is important and essential for visual recognition. Unfortunately, traditional representation learning suffers from fairness issues since the model may learn information of sensitive attributes. Recently, a series of studies have been proposed to improve fairness by explicitly decorrelating target labels and sensitive attributes. Most of these methods, however, rely on the assumption that fully annotated labels on target variable and sensitive attributes are available, which is unrealistic due to the expensive annotation cost. In this paper, we investigate a novel and practical problem of Fair Unsupervised Representation Learning with Partially annotated Sensitive labels (FURL-PS). FURL-PS has two key challenges: 1) how to make full use of the samples that are not annotated with sensitive attributes; 2) how to eliminate bias in the dataset without target labels. To address these challenges, we propose a general Fairness-aware Contrastive Learning (FairCL) framework consisting of two stages. Firstly, we generate contrastive sample pairs, which share the same visual information apart from sensitive attributes, for each instance in the original dataset. In this way, we construct a balanced and unbiased dataset. Then, we execute fair contrastive learning by closing the distance between representations of contrastive sample pairs. Besides, we also propose an unsupervised way to balance the utility and fairness of learned representations by feature reweighting. Extensive experimental results illustrate the effectiveness of our method in terms of fairness and utility, even with very limited sensitive attributes and serious data bias.