What DNS records should I set up for my Namecheap Private Email to work?
Domains hosted with Namecheap can be configured in a way described here

Address Record Type
Here are the types of records that can be created on DNS servers:
A (Address): Allows you to associate a host with an IPv4 address. The IP address that you use does not have to be on your network. For example, you can have the host record for www pointed to 207.46.130.14 (the address for the Microsoft website).
AAAA (IPv6): Allows you to associate a host with an IPv6 address. The IP address that you use does not have to be on your network.
For example, you can have the host record for www pointed to 2001:0db8:11a3:09d7:1f34:8a2e:07a0:765d
MXE (Mail Easy): Allows you to specify the address of your mail server. When you use a mail record, you must use an IP address in the address field.
Note to professional users: Creating a mail record actually creates both the MX and A records in DNS. Also, when using multiple mail servers, a preference value of 10 is used on all entries.
MX (Mail): MX (Mail) can be either a host name under your domain name (for example, mail3), or the name of a mail server (for example, mail.yahoo.com). When using a mail server name, it should end with a period ("."). If you forget to add it, the system will do it automatically.
NOTE: MX records for a domain can be created in the Mail Settings section. For more details, please refer to this article.
CNAME (Alias): Is used to associate a host name with another host. The host that you wish to point to does not have to be on your network.
Please don't set up CNAME for a naked domain (@ hostname) since it may affect the operation of the domain's MX records and, consequently, the email service.
When you use a domain in the address field, you must add a trailing period ("."). Unless you do so, the name server will add your domain to the end of the given domain .
NS (Available on BasicDNS and PremiumDNS only): Are primarily used if you want to divide your domain into subdomains.
Subdomains indicate that you are delegating a portion of a domain name to a different group of nameservers, thus creating NS records to point the hostname (name of the subdomain) to different name servers.
URL Redirect: Standard method for URL Forwarding. When users type in your domain name, they are redirected to the web server that hosts your pages. One drawback of this option is that the browser will display the URL that is on the actual web page, not your domain name.
You can prevent this by using the URL Frame method instead (see below).