How to make 5v DC UPS Circuit?
There are several problems with your circuit:-


A '3.7V' Lithium-Ion cell can be safely be charged up to 4.23V, but no higher or it will explode! Your voltage regulator is set to produce 5.0V, and D2 drops 0.5~0.9V depending on charging current, which could put up to 4.5V on the battery. However D3 drops about 1.7V at normal operating current (assuming it is a red LED, other colors may be higher) leaving only 2.8V so the battery will never get charged.   
Your 5V 3W solar panel should be able to deliver 600mA at 5V in full sunlight. However at 600mA D1 drops ~0.7V and U1 drops ~1.8V, so the regulator output voltage would only be about 2.5V. At low current the solar panel's voltage may rise to ~6.5V and the regulator will have lower voltage drop so its output voltage may be sufficient, but the battery would charge very slowly. 
Most linear regulators require input and output bypass capacitors to prevent unwanted oscillation. The LM317 is better than many, usually only needing 0.1uF across the supply input. If using a different regulator you should follow the manufacturer's recommendations.  


D1 is redundant and should be removed. U1 could be replaced with an LDO regulator such as the MIC29150 which drops less than 0.2V at 600mA.      

You should connect battery negative directly to the solar panel so that charging current doesn't go through the LED. This could result in the battery getting up to 4.5V, so you must reduce the voltage regulator's output voltage until the 'open circuit' voltage after D2 is below 4.23V. 

Your circuit doesn't include over-discharge protection. To prevent the battery from being damaged if for some reason the solar panel cannot maintain sufficient charge, you might consider adding a PCM (Protection Circuit Module). This will also protect against over charging and accidental short circuits.