Most important aspect when you want to lose weight is the diet. Paleo is great, I do paleo / keto myself. To help burn fat and gain muscles, the best thing is to wait about 45-60 minutes before eating (you can drink water), then get some protein. This way you give your body time to replace the glycogen consumed by the training through glyconeogensis, which is a process in which fat and protein is converted into glycogen. In this phase you lose fat and protein (muscles). To keep your muscles it is important to get some protein after the training, especially if you eat paleo / keto since you won't have much glycogen in your blood stream.
Also see this: http://chadwaterbury.com/the-truth-abou ... nutrition/