Probably the most frequently used tool to verify an OC is Prime95. Many people also use OCCT. Typically if a OC runs without errors in Prime 95 or OCCT for around 2 hours it should be stable.
For a safe overclock you want to pay attention to your CPU temps. There are plenty of programs available for you to monitor temps. As far as a safe overclock is concerned, the main things to keep in mind are to keep temps below a certain threshold and voltages low. I don't know the temperatures and voltages that are deemed healthy for a Athlon x2, but as long as your temps and voltages stay below that max temp and voltage you can OC as much as you want.
Obviously using the stock cooler is going to reduce how high you can get your OC before you start to reach temperatures that are unhealthy. If you really want to push your OC you could buy a aftermarket aircooler. They really aren't that expensive, a Hyper 212+ is only ~$30 and is far more effective at keeping temps down compared to the stock cooler.
Some motherboard vendors offer overclocking tools that can be run from the OS. If you want to use such a tool you'll have to check on your motherboard vendor's website. However, I advise against them because they don't choose optimal settings. They frequently set voltages either too low causing instability or too high increasing temps and wear on the CPU more than necessary. I recommend manually overclocking through the BIOS. There are numerous overclocking guides out there that can guide you through the process. Many of them specifically discuss overclocking Athlon CPUs.
Also, I did some research on your CPU and the safe voltage limit appears to be 1.5v and the temp limit appears to be around 60c.
I've used Prime95 and it's pretty simple. Just open the program and go to options->torture test. Set the thread count to 2 and select one of the torture test options click start and it will start running. If it encounters and error it will print a message saying so. If it runs for 2 hours without problems you should be good to go. The first time you run prime95 a dialog box may appear asking you about joining GIMPS, just select the option for testing.
If that doesn't work out you could try OCCT. I've never used it personally, but it's well regarded.
That's what I've heard. You should try to set voltages to the lowest values that are still stable because doing so will reduce heat and wear on the CPU. Also, when torture testing, you want to have a CPU temperature monitoring program running at all times to make sure that CPU temperatures aren't going too high.
The problem with older X2's is that there isn't really much headroom for overclocking, so you'd need to invest in a proper hsf. Even if you did manage to OC it, the % gain isn't really that high (since the OC would be quite limited in the first place). So yeah, it'd probably be better if you leave it as is.