The parameters of the received waveform of a full-waveform satellite laser altimeter can be used for retrieving the morphological information of the target. The traditional waveform processing algorithm is unable to extract the parameters for a non-Gaussian and overlapped multiple-mode waveform. Therefore, a synthetic algorithm with the boosted Richardson-Lucy deconvolution, layered extraction, gradient descent and nonlinear least square was proposed for a skew-normal full-waveform decomposition. The proposed waveform processing experiments were implemented by employing the known-parameter waveforms, airborne simulated waveforms and global ecosystem dynamics investigation (GEDI) lidar waveforms and the evaluation indictors including the waveform correlation coefficient, root mean square error (RMSE), relative error of characteristic parameters, successful detection rate of the number of components. The processed results were compared with those by the traditional Gaussian decomposition algorithm. The average correlation coefficient of the processing results for the known-parameter data set had a growth of 2% and the average RMSE has a reduction of 47%. The average relative error of parameters was reduced by about 5% and successful detection rate of the number of components was improved by about 34%. For the simulated and GEDI lidar waveforms, the average correlation coefficients had slight growth of 1% and 2%, and the average RMSEs had a more significant reduction of 56% and 54%, respectively. In addition, the simulated verification of canopy height in steeped region was carried out. The precision of the derived canopy height was significantly higher than that of the traditional method. All processed results demonstrate that the proposed method is more conducive to the extraction of the multiple-mode waveform parameters and the inversion of target parameters.