[Client] 채팅기능 구현 - 렌더링 작업

### ISSUE
- Group: `client`
- Type: `react`
- Detail: 채팅기능 렌더링 작업 진행하기

### TODO
1. [ ] 렌더링 작업 진행
2. [ ] 서버와 연결 후 작동 여부 체크하기

### Estimated time
> Pick one
### `3h`

### Labels
- Estimated time: `E: 3h`
- Group : `client`
- Sprint: `Sprint__2`
- Urgency: `High`

Start by locating the client chat feature and reviewing its existing rendering flow, then identify the server connection path. Done means the chat UI renders correctly and its operation is checked after connecting to the server.

Written by the indexing model from the issue text.