It's OK. I ride it often, both directions, but it does need great caution going down. The surface can be a bit sketchy. Show it due respect when descending and you'll be fine.
Alternative hills around there are - West to East - Gangers Hill (a bit easier) Tandridge Hill (Similar to Chalkpit) Chalkpit La, Titsey Hill (more of a main rd, better surface but busy) White La, (very steep, narrow branch off Titsey Hill).
Easiest ascent/descent round there is Clarks Lane. Wide, good visibility, not too steep, decent surface, moderately busy.
Further East you have the main road to Biggin Hill which I don't cycle as it's too busy with fast traffic, but my little car struggles on that hill, second gear needed. Then you have the triumvirate of Hogtrough, Brasted and Sundridge hills, all of which are steep.