How to use a variable in a CURL request with bash?

Using jq for this, as Charles Duffy's answer suggests, is a very good idea. However, if you can't or do not want to install jq here is what you can do with plain POSIX shell.

#!/bin/sh
set -e

current_ip="$(curl --silent --show-error --fail ipecho.net/plain)"
echo "IP: $current_ip"

# Update A record
curl -X PUT "https://api.cloudflare.com/client/v4/zones/ZONEIDHERE/dns_records/DNSRECORDHERE" \
    -H "X-Auth-Email: EMAILHERE" \
    -H "X-Auth-Key: AUTHKEYHERE" \
    -H "Content-Type: application/json" \
    --data @- <<END;
{
    "id": "ZONEIDHERE",
    "type": "A",
    "name": "example.com",
    "content": "$current_ip",
    "zone_name": "example.com"
}
END

The reliable way to edit JSON from shell scripts is to use jq:

# set shell variables with your contents
email="yourEmail"
authKey="yourAuthKey"
zoneid="yourZoneId"
dnsrecord="yourDnsRecord"

# make sure we show errors; --silent without --show-error can mask problems.
current_ip=$(curl --fail -sS ipecho.net/plain) || exit

# optional: template w/ JSON content that won't change
json_template='{"type": "A", "name": "example.com"}'

# build JSON with content that *can* change with jq
json_data=$(jq --arg zoneid "$zoneid" \
               --arg current_ip "$current_ip" \
               '.id=$zoneid | .content=$current_ip' \
               <<<"$json_template")

# ...and submit
curl -X PUT "https://api.cloudflare.com/client/v4/zones/$zoneid/dns_records/$dnsrecord" \
  -H "X-Auth-Email: $email" \
  -H "X-Auth-Key: $authKey" \
  -H "Content-Type: application/json" \
  --data "$json_data"