For your first day, you can just walk to Tiananmen Square and the forbidden City. The hotel is just next to Wanfujing street. A bit tight in second day. You should get up earlier in the morning to take 919 bus at Deshengmen Gate to Badaling. You can take bus 82 at Nanheyan Station just next to your hotel and get off at Gulou (Drum Tower). Then change to Bus 5 and get off at Deshengmen Station. Find the Deshengmen Arrow Tower. The 919 bus is just behind the tower. The bus takes nearly a hour if the traffic is all right. If you arrive in Badaling before 10am, spend 3-5 hour in the Great Wall. It will be the afternoon when you get back to Deshengmen. Then you have to take subway (Line 2 from Jishuitan to Line 5 to Temple of Heaven). In temple of Heaven, stop sell ticket at 16:00, so you had better get there no later than 15:00. That means you should be back to Deshengmen from the Great Wall around 14:00. Anyway, arrange your time.
Day 3, Walk to Tiananmen East Subway Station and take Line 1 and change to Line 4 to Beiongmen Station to Summer Palace. In Beigongmen (north gate of Summer Palace), there is direct bus to xiangshan. The lying Buddha Temple is close to Xiangshan. I suggest you give up the lying buddha.Time is not enough.
Day 4, do you mean Shichahai by chuchuchai? You can take bus 82 to Drumer Tower, then walk around Shichahai, then North Gate of Beihai Park. Get out from the south gate and take bus to Xidan, not far away. Then Take Subway Line 4 from Xidan to Haidianhuangzhuang. Change to Line 10 to Beitucheng and then change to Line 8 to Olympic Center. Or you can pur Xidan after then Bird's nest and Water Cube.
Day 5, no time for World Park, I guess. Temple of Confucious is together with Guozijian and Yonghe Monastery. I suggest you go there.

many thanks

Mr. Qi, thanks, in day 4 - you did not mention fragrant Hills Park, and in day 5 - what do you mean by "no time for world park" is that because world park is too far away?

Hi, Frangrant Hill is in your plan in Day 3 and Xiangshan I mentioned is just the Fragrant Hill. Yes, World Park is far from other places in Day 5 and compared with Yonghe Monastery, it is less worth visiting.