How to... extend my visa in Beijing

Your safe haven for extensions of L and F visa is the PSB or Exit and Entry Management Section of the Public Security Bureau.
Concerning the extension of L (tourist) visa, make sure you check the latest regulations mentioned in the PSB . Please be aware that restrictions have been tightened and most applicants will need to prove they have sufficient funds on their own Chinese bank account (sufficient funds being defined as 100$/day of stay in China or 25.000 RMB for a 30 days extension). According to current information, tourist visa can be extended twice in Beijing for 30 days, up to a total duration of stay of 90 days. However, stricter rules apply to nationals of other Asian countries, who may only be able to extend theuir visa once for ten days.

When it comes to finding the perfect timing strategy, be advised that as long as your passport lands on a PSB desk the day your visa expires, you're completely fine. Your new visa will automatically be issued from the day you hand your documents in. If you want to be 120% sure, go the day before its expiry so that you can come back in case you forgot a document.

The Exit&Entry Section is located at
2 Andingmen Dongdajie, Dongcheng District
(southeast of Beixiaojie Bridge)

If you take public transportation, you can take
Bus number 44, 13, 116, 117, 807 to Beixiaojie Huokou or
Subway Line 2 to Lama Temple (Yonghegong) station.
If you're going by taxi, you might want to print out their Chinese address:

公安局出入境管理处
东城区安定门东大街2号（北小街桥东南）

A normal extension should take about five working days, but do not forget to consider potential public holidays.
If you need to travel within China while your passport is at the PSB, you can do so with the receipt slip (or "domestic travel authorization") they give you when you hand in your documents. However, be advised that especially in smaller Chinese cities this may lead to confusion at the airport, resulting in longer processing times. I therefore advice you to be at the airport a bit earlier and bring your alien employment permit if you have one.

Travelling outside of China, including to Hong Kong, is not possible without your passport.

The PSB's phone numbers are
(0086 10) 84020101 for the information desk;
(0086 10) 84015316 for the general customer service line.